#HNBO 3.0 Program Manual#N #N(#INatural Bond Orbital / Natural Population Analysis / Natural Localized Molecular Orbital Programs#N) E. D. Glendening, A. E. Reed,9agger J. E. Carpenter,9dagger and F. Weinhold #ITheoretical Chemistry Institute and Department of Chemis- try, University of Wisconsin, Madison, Wisconsin 53706#N 92717.#N 0 The various natural localized sets can be considered to result from a sequence of transformations of the in- put atomic orbital basis set { _______________ 9agger #C#RNote, however, that some electronic structure packages do not make provision for calculating the spin density matrices for some types of open-shell wavefunc- tions (e.g., MCSCF wavefunctions calculated by the GUGA formalism in the GAMESS system), so that NBO analysis cannot be applied in these cases.#O 9dagger #C#RIf the wavefunction is not calculated in an atom-centered basis set, it would be necessary to first compute a wavefunction for each isolated atom of the molecule (in the actual basis set and geometry of the molecular calculation), then select the most highly oc- cupied natural orbitals of each atomic wavefunction to compose a final set of linearly independent atom- centered basis functions of the required dimensionali- ty. Since atom-centered basis functions are the nearly universal choice for molecular calculations, the current _________________________ July 11, 1995 - 2 - NBO program makes no provision for this step.#O Guide, Section C. +0 +lf 0>> +0 +lf 0//' |<<3//15//75//8//35>>|7____________________99____________________ -dwid +0 ipen //+0 +dht ipen //+dhlfwid +0 ipen >> input basis7 arr NAOs7 arr NHOs7 arr NBOs7 arr NLMOs _________________________ Each natural localized set forms a complete orthonormal set of one-electron functions for expanding the delo- calized molecular orbitals (MOs) or forming matrix representations of one-electron operators. The overlap of associated ``pre-orthogonal'' NAOs (PNAOs), lacking only the interatomic orthogonalization step of the NAO procedure, can be used to estimate the strength of or- bital interactions in the usual way. 0 The optimal condensation of occupancy in the natural localized orbitals leads to partitioning into high- and low-occupancy orbital types (reduction in dimensionali- ty of the orbitals having significant occupancy), as reflected in the orbital labelling. The small set of most highly-occupied NAOs, having a close correspon- dence with the effective minimal basis set of semi- empirical quantum chemistry, is referred to as the ``natural minimal basis'' (NMB) set. The NMB (core + valence) functions are distinguished from the weakly occupied ``Rydberg'' (extra-valence-shell) functions that complete the span of the NAO space, but typically make little contribution to molecular properties. Similarly in the NBO space, the highly occupied NBOs of the natural Lewis structure can be distinguished from the ``non-Lewis'' antibond and Rydberg orbitals that complete the span of the NBO space. Each pair of valence hybrids , in the NHO basis give rise to a bond (b ) and antibond (ab ) in the NBO basis, b = ab = the former a Lewis (L) and the latter a non-Lewis (NL) orbital. The antibonds (valence shell non-Lewis orbi- tals) typically play the primary role in departures (delocalization) from the idealized Lewis structure. 0 The NBO program also makes extensive provision for energetic analysis of NBO interactions, based on the availability of a 1-electron effective energy operator (Fock matrix) for the system. Estimates of energy ef- fects are based on second-order perturbation theory, or on the effect of deleting certain orbitals or matrix elements and recalculating the total energy. NBO ener- gy analysis is dependent on the specific ESS to which the NBO program is attached, as described in the Appen- July 11, 1995 - 3 - F x y t 0.0001 0.5 .05 dpx dpy 0 1 ipen //+radius +0 0// F x ym t 0.0001 0.5 .05 dpx dpy 0 1 ipen // +0 +dht ipen //+diam -dht 0//+0 +dht ipen //-radius +0 0// F x ym t 0.0001 -0.5 -0.05 dpx dpy 0 1 ipen // +radius -dhlfht 0//'ext _________________________ dix. 0 The program is provided in a core set of NBO routines that can be attached to an electronic structure system of the user's choice. In addition, specific `driver' routines are provided that facilitate the attachment to popular #Iab initio#N and semi-empirical packages (GAUSSIAN-8X, GAMESS, HONDO, AMPAC, etc.). These ver- sions are described in individual Appendices. #IA.1.2 #IStructure of the NBO Program#N 0 The overall logical structure of the NBO program and its attachment to an electronic structure system (ESS) are illustrated in the block diagram, Fig. 1. This figure illustrates how the ESS and its scratch files (in the upper part of the diagram) communicate through the interface routines RUNNBO, FEAOIN, and DELSCF with the main NBO modules and associated direct access file (in the lower part). 0 The main NBO program is represented by modules la- belled ``NBO'' and ``NBOEAN.'' These refer to the con- struction of NBOs (including natural population analysis, construction of NAOs, NLMOs, etc.) and to NBO energy analysis, respectively. Each module consists of subroutines and functions that perform the required operations. These two modules communicate with the direct-access disk file NBODAF (LFN 48, labelled ``FILE48'' elsewhere in this manual) that is created and maintained by the NBO routines. Details of the NBO and NBOEAN modules, common blocks, and direct- access file are described in the Programmer's Guide, Section C. 0 The NBO program blocks communicate with the attached ESS through three system-dependent `driver' subroutines (RUNNBO, FEAOIN, DELSCF). The purpose of these drivers is to load needed information about the wavefunction and various matrices into the FILE48 direct access file and NBO common blocks. Although the ESS is usually thought of as `driving' the NBO program, from the point of view of the NBO program the ESS is merely a `device' that provides initial input (e.g., a density matrix and label information) or other feedback (a calculated en- ergy value) upon request. Each such ESS device there- fore requires special drivers to make this feedback July 11, 1995 - 4 - ' 2//+0 +lf 0//'file' 2>> #BFigure 1:#N Schematic diagram depicting flow of informa- tion between the electronic structure system (ESS) and the NBO program, and the commun#|ication lines connecting these programs _________________________ possible. Versions of the driver subroutines are in- cluded for several popular packages. The driver rou- tines are described in more detail in the Programmer's 9 agger #RPresent address: Bayer AG, Abteilung AV-IM-AM, 5090 Leverkusen, Bayerwerk, Federal Republic of Ger- many. 9 dagger #RPresent address: Department of Chemistry, University of California-Irvine, Irvine, California #HTable of Contents#N #HPREFACE: HOW TO USE THIS MANUAL#N 0 The NBO manual is divided into three major sections: 0 Section A (``General Introduction and Installation'') contains general introductory and `one-time' informa- tion for the novice user: what the program does, pro- gram structure and relationship to driver electronic structure package, initial installation, `quick start' sample input data, and a brief tutorial on sample out- put. 0 Section B (``NBO User's Guide'') is for the inter- mediate user who has an installed program and general familiarity with the standard (default) options of the NBO program. This section documents the list of #Ikeywords#N that can be used to alter the standard NBO job options, with examples of the resulting output. This section is mandatory for users who wish to use the July 11, 1995 - 5 - to the ESS scratch file (called the ``dictionary file,'' ``read-write file,'' etc., in various systems) and the NBO direct access file (NBODAF). Heavier box borders mark the ESS-specific driver routines (RUNNBO, FEAOIN, _________________________ program to its full potential, to `turn off' or `turn on' various NBO options for their specialized applica- tions. 0 Section C (``NBO Programmer's Guide'') is for accom- plished programmers who are interested in program logic and the detailed layout of the source code. This sec- tion describes the relationship of the source code sub- programs to the published algorithms for NAO, NBO, and NLMO determination, providing documentation at the lev- el of individual common blocks, functions, and subrou- tines. This in turn serves as a bridge to the `micro- documentation' included as comment statements within the source code. Section C also provides guidelines for constructing `driver' routines to attach the NBO programs to new electronic structure packages. #HSection A: GENERAL INTRODUCTION AND INSTALLATION#N #BA.1 INTRODUCTION TO THE NBO PROGRAM#N #IA.1.1 #IWhat Does the NBO Program Do?#N 0 The NBO program performs the analysis of a many- electron molecular wavefunction in terms of localized electron-pair `bonding' units. The program carries out the determination of natural atomic orbitals (NAOs), natural hybrid orbitals (NHOs), natural bond orbitals (NBOs), and natural localized molecular orbitals (NLMOs), and uses these to perform natural population analysis (NPA), NBO energetic analysis, and other tasks pertaining to localized analysis of wavefunction pro- perties. The NBO method makes use of only the first- order reduced density matrix of the wavefunction, and hence is applicable to wavefunctions of general mathematical form; in the open-shell case, the analysis is performed in terms of ``different NBOs for different spins,'' based on distinct density matrices for lpha and777777 t99a999999 s99p99i99n99.99999999999999999a99g99g777777r This section provides a brief introduction to NBO algorithms and nomenclature. 0 NBO analysis is based on a method for optimally transforming a given wavefunction into localized form, corresponding to the one-center (``lone pair'') and two-center (``bond'') elements of the chemist's Lewis structure picture. The NBOs are obtained as local block eigenfunctions of the one-electron density ma- July 11, 1995 - 6 - DELSCF) that directly interface the ESS program. The heavy dashed lines denote calls from the NBO program `backward' to the ESS program for information needed to carry out its tasks. Otherwise, the sequential flow of program control is generally from top to bottom and from left to right in the diagram. #IA.1.3 #IInput and Output#N 0 From the user's point of view, the #_input#/ to the NBO program attached to an ESS program consists simply of one or more keywords (an NBO #Ikeylist#N) included in the ESS input file. In effect, the NBO program reads these keywords to set various job options, then interrogates the ESS program through the DELSCF and FEAOIN drivers for additional information concerning the wavefunction. The general form of NBO keylists and the specific functions associated with each keyword are detailed in the User's Guide, Section B. The method of including NBO keylists in the input file for each ESS is detailed in the specific Appendix for the ESS. 0 The following information is passed from the ESS to the NBO program (transparent to the user): The one-electron density matrix #BD#N (or density matrices in the open-shell case) in the chosen atomic orbital (AO) basis set; The AO overlap matrix #BS#N, and label information identifying the symmetry (angular momentum type) and location (number of the atom to which affixed) for each AO; Atomic number (nuclear charge) of each atom. Certain additional information is written on the FILE48 direct access file and may be used in response to specific job options, such as the AO Fock matrix #BF#N, if energy analysis is requested; the AO dipole matrix #BM#N, if dipole moment analysis is requested; or information concerning the mathematical form of the AOs (orbital exponents, contraction coefficients, etc.), if orbital _________________________ trix, and are hence ``natural'' in the sense of L8mlaut owdin, having optimal convergence properties for describing the electron density. The set of high- occupancy NBOs, each taken doubly occupied, is said to represent the ``natural Lewis structure'' of the molecule. Delocalization effects appear as weak depar- tures from this idealized localized picture. July 11, 1995 - 7 - plotting information is requested to be saved as input for a contour plotting program. 0 The principal #_output#/ from the NBO program con- sists of the tables and summaries describing the results of NBO analysis, included in the ESS output file. Sample NBO output is described in Section A.2.4 below. If requested, the NBO program may also write out transformation matrices or other data to disk files. The NBO program also creates or updates two files, the direct-access file (FILE48) and the `archive' file (FILE47) that can be used to repeat NBO analysis with different options, without running the ESS program to recalculate the wavefunction. Necessary details of these files are given in Section B.7 and the Programmer's Guide, Section C. #IA.1.4 #IGeneral Capabilities and Restrictions#N 0 Principal capabilities of the NBO program are: Natural population, natural bond orbital, and natural localized molecular orbital analysis of SCF, MCSCF, CI, and M0t oller-Plesset wavefunctions (main subroutine: NBO); For RHF closed-shell and UHF wavefunctions only, ener- getic analysis of the wavefunction in terms of the interactions (Fock matrix elements) between NBOs (main subroutine: NBOEAN); Localized analysis of molecular dipole moment in terms of NLMO and NBO bond moments and their interactions (main subroutine: DIPANL). 0 A highly transportable subset of standard FORTRAN 77 is employed, with no special compiler extensions of any vendor, and all variable names of six characters or less. Common abbreviations used in naming subprograms, variables, and keywords are: = overlap matrix = density matrix (or D) = Fock matrix = dipole matrix (or DXYZ, or DX, DY, DZ) = Natural Population Analysis = Natural Atomic Orbital = Natural Bond Orbital = Natural Localized Molecular Orbital = pre-orthogonal NAO (i.e., omit interatomic orthogonali- zation) hsp 0 Most of the NBO storage is allocated dynamically, to conform to the minimum required for the molecular sys- tem under study. However, certain NBO common blocks of fixed dimensionality are used for integer storage. These are currently dimensioned to accomodate up to 99 July 11, 1995 - 8 - atoms and 500 basis functions. Section C.3 describes how these restrictions can be altered. The program is not set up to handle complex wavefunctions, but can treat any real RHF, ROHF, UHF, MCSCF (including GVB), CI, or M0t oller-Plesset-type wavefunction (i.e., any form of wavefunction for which the requisite density matrices are available) for ground or excited states of general open- or closed-shell molecules. Effective core poten- tials (``pseudo#|potentials'') can be handled, includ- ing complete neglect of core electrons as assumed in semi-empirical treatments. The atomic orbital basis functions (up to #If#N orbitals in angular symmetry) may be of general Slater-type, contracted Gaussian- type, or other general composition, including the ``effective'' ortho#|normal valence-shell AOs of semi- empirical treatments. AO basis functions are assumed to be normalized, but in general non-orthogonal. #IA.1.5 #IReferences and Relationship to Previous Versions#N 0 This program (``version 3.0'') is an extension of previous versions of the NBO method incorporated in the semi-empirical program #IBONDO#N [F. Weinhold, #IQuan- tum Chemistry Program Exchange No. 408#N (1980); ``ver- sion 1.0''] and in a GAUSSIAN-82 implementation [A. E. Reed and F. Weinhold, #IQCPE Bull. #B5#N, 141 (1985); ``version 2.0''], and should be considered to supplant those versions. Version 3.0 also supplants the various specific versions (``the GAMESS version,'' ``the AMPAC version,'' etc.) that have been informally created and distributed to individual users outside the QCPE frame- work. Principal contributors to the development of the NBO methods and programs (1975-1990) are Principal references to the development and applica- tions of NAO/NBO/NLMO methods are: J. P. Foster and F. Weinhold, #IJ. Am. Chem. Soc. #B102#N, 7211-7218 (1980). A. E. Reed and F. Weinhold, #IJ. Chem. Phys. #B78#N, 4066-4073 (1983); A. E. Reed, R. B. Weinstock, and F. Weinhold, #IJ. Chem. Phys. #B83#N, 735-746 (1985). A. E. Reed and F. Weinhold, #IJ. Chem. Phys. #B83#N, July 11, 1995 - 9 - 1736-1740 (1985). J. E. Carpenter and F. Weinhold, #IJ. Molec. Struct. (Theochem) #B169#N, 41-62 (1988); J. E. Carpenter, #IPh. D. Thesis#N, University of Wisconsin, Madison, 1987. A. E. Reed, L. A. Curtiss, and F. Weinhold, #IChem. Rev. #B88#N, 899-926 (1988); F. Weinhold and J. E. Car- penter, in, R. Naaman and Z. Vager (eds.), ``The Struc- ture of Small Molecules and Ions,'' (Plenum, New York, 1988), pp. 227-236. 0 The principal enhancements of version 3.0 include: #IGeneralized Program Interface.#N Overall program organization (Fig. 1) has been modified to standardize communication with the main ESS program. This insures that all special ESS ``versions'' of the NBO program now have consistent options and capabilities (as long as the option is meaningful in the context of the ESS), and enables the program to be offered in a greater number of specialized ESS versions than were previously available. #INAO/NPA Summary Table.#N New tables give improved display of NAOs and natural populations, including the ``natural electron configuration'' of each atom (i.e., the occupancy and type of NAOs describing the atomic electron configuration of each atom). The new NAO sum- mary tables (Section A.3.2) include an SCF atomic orbi- tal energy (if available), a conventional atomic orbi- tal label (1#Is#N, 2#Is#N, 2#Ip#N, etc., in accordance with the labelling in isolated atoms), and a shell designation (Cor = core, Val = valence, or Ryd = Ryd- berg) to aid characterization of the NAO. #INBO Summary Table.#N A new NBO summary table (Section A.3.6) has been provided to summarize the energetics and delocalization patterns of the principal NBOs. This succinctly combines the most important information from the full NBO table, diagonal NBO Fock matrix ele- ments, and 2nd-order energy analysis. #IBond Bending Analysis.#N The program includes a new analysis of hydrid directionality and bond ``bending'' (keyword BEND, Section A.3.4). #IDipole Moment Analysis.#N The program includes new optional provision (keyword DIPOLE, Section B.6.3) for analysis of the molecular dipole moment in terms of July 11, 1995 - 10 - localized NLMOs and NBOs. #IPrint options.#N The program offers new structured printing options (Section B.2.4) that give greater con- venience and flexibility in controlling printed output, with improved provision for printing matrices or basis transformations involving general NAO, NHO, NBO, NLMO or pre-orthogonal (PNAO, PNHO, PNBO, PNLMO) basis sets. #IOrbital Contour Info.#N The program makes optional provision (keyword PLOT, Section B.2.5) for writing out files that can be used by an orbital plotting program (available separately through QCPE) to draw contour diagrams of the NBOs or other natural localized orbi- tals. #IEffective Core Potentials.#N The program now handles effective core potentials (pseudo#|potentials), or the complete neglect of core levels characteristic of semi-empirical wavefunctions (Section B.6.12). The program also includes three changes to correct problems of the previous version (which may have affected a small number of users): #IUnpolarized Cores.#N NAOs identified as ``core'' orbitals are now auto#|matically carried over as unhy- bridized 1-center core NBOs (Section B.3). This has virtually no effect on the form or occupancy of a core NBO, but averts the (rare) problem of unphysical mixing between core and valence lone pairs when the occupan- cies are `accidentally' degenerate (usually, both very close to 2.000...) within the numerical machine preci- sion. A warning message is printed when the core occu- pancy is less than 1.9990, indicating a possible ``core polarization'' effect of physical significance. #IExcited State Antibond Labels.#N The program now directly investigates the nodal structure of an NBO (by examining the overlap matrix in the PNHO basis) before assigning it a label as a ``bond'' (unstarred) or ``antibond'' (starred) NBO. In previous versions, these labels were assigned on the basis of the presumed higher occupancy of the in-phase bond combination, which was generally true for ground states, but not for excited states. The program now prints a warning mes- sage whenever it encounters the ``anomalous'' situation of an out-of-phase antibond NBO having higher occupancy than the corresponding in-phase bond NBO, indicative of an excited-state configuration. [WARNING: the overlap test cannot be applied to semi-empirical methods with orthogonal AOs (e.g., AMPAC), so antibond labels for these methods are assigned, as in previous versions, on July 11, 1995 - 11 - the basis of occupancy.] #IAlternative Resonance Structures.#N The program now institutes a search for alternative Lewis (`resonance') structures when two or more structures may be competi- tive, and returns the structure of lowest non-Lewis occupancy. This corrects a possible dependence on atomic numbering in cases of strong delocalization. Despite these changes and extensions, version 3.0 has been designed to be upward compatible with v. 2.0, as nearly as possible. Previous users of NBO 2.0 should find that their jobs run similarly (i.e., most keywords continue to function as in previous versions). Thus, experienced NBO users should find little difficulty in adapting to, and experimenting with, the new capabili- ties of the program. #BA.2 INSTALLING THE NBO PROGRAM#N 0 The NBO programs and manual are provided on a distri- bution tape. The tape contains three files: the TechSet code of this manual (file NBO.MAN), a file con- taining the core NBO source routines and supporting driver routines (file NBO.SRC), and the Fortran ``ena- bler'' program (file ENABLE.FOR). 0 In overview, the installation procedure involves the following steps (the details of each step being depen- dent on your operating system): #IEnabling the NBO routines.#N Copy the contents of the distribution tape onto your system. Using your system Fortran 77 compiler, compile and link the ena- bler program to create the ENABLE.EXE executable; for example, the VMS commands to create ENABLE.EXE are #T FOR ENABLE LINK ENABLE #NNow, run the ENABLE program (e.g., type ``RUN ENABLE'' in a VMS system), and answer the prompt #T NBO program version to enable? #Nby selecting from the available offerings. Each ESS package is associated with a 3-letter identifier (``G88'' for GAUSSIAN-88, ``GMS'' for GAMESS, ``AMP'' for AMPAC, etc.). The ENABLE program will create a file #IXXX#NNBO.FOR (where `#IXXX#N' is the identifier) that incorporates the appropriate drivers for your ESS. July 11, 1995 - 12 - #ICompiling the NBO routines.#N Using your system For- tran 77 compiler, compile the #IXXX#NNBO.FOR file to an object code file (say, #IXXX#NNBO.OBJ). [Compiler errors (if any) should be fixed before proceeding. Please notify the authors if you encounter undue diffi- culties in this step.] #IModifying the ESS routines.#N In general, the ESS source Fortran code must be modified to call the NBO routines near the point where the ESS performs Mulliken Population Analysis or evaluates properties of the final wavefunction. The modification generally con- sists of inserting a single statement (viz., ``CALL RUNNBO'') in one subroutine of your ESS system. See the appropriate Appendix of this Manual for detailed information on exactly how to modify the ESS code for your chosen system. #IRebuilding the integrated ESS/NBO program.#N Re- compile your modified ESS programs and link the result- ing object file (say, ESS.OBJ) with the #IXXX#NNBO.OBJ file to form the final ESS.EXE executable. In general, this step will closely follow the initial installation procedure for your ESS, with the exception that the #IXXX#NNBO.OBJ file must be included in the link state- ment (or deposited in one of the libraries accessed by the linker, etc.). Note that installation of the NBO programs into your ESS system in no way affects the way your system processes standard input files. The only change involves enabling the reading of NBO keylists (if detected in your input file), performance of the tasks requested in the keylist, and return of control to the parent ESS program in the state in which the NBO call was encountered. 0 If you are interfacing the NBO programs to a new ESS package (not represented in the driver routines pro- vided with this distribution), see Section C for gui- dance on how to create drivers for your ESS to provide the necessary information. Alternatively, see Section B.7 for a description of the input file to GENNBO, the stand-alone version of the NBO program. 0 The TechSet-coded version of this manual, NBO.MAN, can be printed on an HP LaserJet printer (`F' car- tridge) with the TECHSET technical typesetting program [ACS Software, American Chemical Society, Marketing Communications Dept., 1155 Sixteenth Street, N.W., Washington, D.C. 20036]. #BA.3 TUTORIAL EXAMPLE FOR METHYLAMINE#N July 11, 1995 - 13 - #IA.3.1 Running the Example#N 0 This section provides an introductory `quick start' tutorial on running a simple NBO job and interpreting the output. The example chosen is that of methylamine (CH#d3#uNH#d2#u) in Pople-Gordon idealized geometry, treated at the #Iab initio#N RHF/3-21G level. This simple split-valence basis set consists of 28 AOs (nine each on C and N, two on each H), extended by 13 AOs beyond the minimal basis level. 0 Input files to run this job (or its nearest equivalent) with each ESS are given in the Appendix. (The output shown below was created with the GAMESS system.) In most cases, you can modify the standard ESS input file to produce NBO output by simply includ- ing the line #T $NBO $END #Nat the end of the file. This is an `empty' NBO keyl- ist, specifying that NBO analysis should be carried out at the #Idefault#N level. 0 The default NBO output produced by this example is shown below, just as it appears in your output file. The start of the NBO section is marked by a standard header and storage info: ******************************************************************************* N A T U R A L A T O M I C O R B I T A L A N D N A T U R A L B O N D O R B I T A L A N A L Y S I S ******************************************************************************* Job title: Methylamine...RHF/3-21G//Pople-Gordon stan- dard geometry Storage needed: 2505 in NPA, 2569 in NBO ( 750000 available) #T @seg #NNote that all NBO output is formatted to a maximum 80-character width for convenient display on a computer terminal. The NBO heading echoes any requested key- words (none for the present default case) and shows an estimate of the memory requirements (in double preci- sion words) for the separate steps of the NBO process, compared to the total allocated memory available through your ESS process. Increase the memory allo- cated to your ESS process if the estimated NBO requests exceed the available storage. #IA.3.2 Natural July 11, 1995 - 14 - Population Analysis#N #N0 The next four NBO output segments summarize the results of natural population analysis (NPA). The first segment is the main NAO table, as shown below: NATURAL POPULATIONS: Natural atomic orbital occupan- cies NAO Atom # lang Type(AO) Occupancy Energy ---------- ----------------------------------------------- 1 C 1 s Cor( 1s) 1.99900 -11.04184 2 C 1 s Val( 2s) 1.09038 -0.28186 3 C 1 s Ryd( 3s) 0.00068 1.95506 4 C 1 px Val( 2p) 0.89085 -0.01645 5 C 1 px Ryd( 3p) 0.00137 0.93125 6 C 1 py Val( 2p) 1.21211 -0.07191 7 C 1 py Ryd( 3p) 0.00068 1.03027 8 C 1 pz Val( 2p) 1.24514 -0.08862 9 C 1 pz Ryd( 3p) 0.00057 1.01801 10 N 2 s Cor( 1s) 1.99953 -15.25950 11 N 2 s Val( 2s) 1.42608 -0.71700 12 N 2 s Ryd( 3s) 0.00016 2.75771 13 N 2 px Val( 2p) 1.28262 -0.18042 14 N 2 px Ryd( 3p) 0.00109 1.57018 15 N 2 py Val( 2p) 1.83295 -0.33858 16 N 2 py Ryd( 3p) 0.00190 1.48447 17 N 2 pz Val( 2p) 1.35214 -0.19175 18 N 2 pz Ryd( 3p) 0.00069 1.59492 19 H 3 s Val( 1s) 0.81453 0.13283 20 H 3 s Ryd( 2s) 0.00177 0.95067 21 H 4 s Val( 1s) 0.78192 0.15354 22 H 4 s Ryd( 2s) 0.00096 0.94521 23 H 5 s Val( 1s) 0.78192 0.15354 24 H 5 s Ryd( 2s) 0.00096 0.94521 25 H 6 s Val( 1s) 0.63879 0.20572 26 H 6 s Ryd( 2s) 0.00122 0.99883 27 H 7 s Val( 1s) 0.63879 0.20572 28 H 7 s Ryd( 2s) 0.00122 0.99883 #T @seg #NFor each of the 28 NAO functions, this table lists the atom to which NAO is attached (in the numbering scheme of the ESS program), the angular July 11, 1995 - 15 - momentum type `lang' (#Is#N, #Ip#dx#u#N, etc., in the coor- dinate system of the ESS program), the orbital type (whether core, valence, or Rydberg, and a conventional hydrogenic-type label), the orbital occupancy (number of electrons, or `natural popu- lation' of the orbital), and the orbital energy (in the favored units of the ESS program, in this case atomic units: 1 a.u. = 627.5 kcal/mol). [For example, NAO 4 (the highest energy C orbital of the NMB set) is the valence shell 2#Ip#N#dx#u orbital on carbon, occupied by 0.8909 electrons, whereas NAO 5 is a Rydberg 3#Ip#N#dx#u orbital with only 0.0014 electrons.] Note that the occu- pancies of the Rydberg (Ryd) NAOs are typically much lower than those of the core (Cor) plus valence (Val) NAOs of the natural minimum basis set, reflecting the dominant role of the NMB orbi- tals in describing molecular properties. 0 The principal quantum numbers for the NAO labels (1#Is#N, 2#Is#N, 3#Is#N, etc.) are assigned on the basis of the energy order if a Fock matrix is available, or on the basis of occupancy otherwise. A message is printed warning of a `population inversion' if the occupancy and energy ordering do not coincide. Summary of Natural Population Analysis: Natural Population Natural --------- -------------------------------------- Atom # Charge Core Valence Rydberg Total ---------- ------------------------------------------------------------- C 1 -0.44079 1.99900 4.43848 0.00331 6.44079 N 2 -0.89715 1.99953 5.89378 0.00384 7.89715 H 3 0.18370 0.00000 0.81453 0.00177 0.81630 H 4 0.21713 0.00000 0.78192 0.00096 0.78287 H 5 0.21713 0.00000 0.78192 0.00096 0.78287 H 6 0.35999 0.00000 0.63879 0.00122 0.64001 H 7 0.35999 0.00000 0.63879 0.00122 0.64001 ======================================================================= * Total * 0.00000 3.99853 13.98820 0.01328 18.00000 July 11, 1995 - 16 - #NThe next segment is an atomic summary showing the natural atomic charges (nuclear charge minus summed natural popula- tions of NAOs on the atom) and total core, valence, and Ryd- berg populations on each atom: #T @seg #NThis table succinctly describes the molecular charge distribution in terms of NPA charges. [For example, the carbon atom of methylamine is assigned a net NPA charge of minus 0.441 at this level; note also the slightly less positive charge on H(3) than on the other two methyl hydrogens: +0.184 vs. +0.217.] Natural Population -------- ------------------------------------------------ Core 3.99853 ( 99.9632% of 4) Valence 13.98820 ( 99.9157% of 14) Natural Minimal Basis 17.98672 ( 99.9262% of 18) Natural Rydberg Basis 0.01328 ( 0.0738% of 18) --- ----------------------------------------------------- #NNext follows a summary of the NMB and NRB populations for the composite system, summed over atoms: #T @seg #NThis exhibits the high percentage contribution (typically, > 99%) of the NMB set to the molecular charge distribution. [In the present case, for example, the 13 Rydberg orbitals of the NRB set contribute only 0.07% of the electron den- sity, whereas the 15 NMB functions account for 99.93% of the total.] #NFinally, the natural populations are summarized as an effective valence electron configuration (``natural electron configuration'') for each atom: Atom # Natural Electron Configuration --------- - ------------------------------------------------------------------ C 1 [core]2s( 1.09)2p( 3.35) N 2 [core]2s( 1.43)2p( 4.47) H 3 1s( 0.81) H 4 1s( 0.78) H 5 1s( 0.78) H 6 1s( 0.64) H 7 1s( 0.64) #T @seg July 11, 1995 - 17 - #NAlthough the occupancies of the atomic orbitals are non-integer in the molecular environment, the effective atomic configurations can be related to idealized atomic states in `promoted' configura- tions. [For example, the carbon atom in the above table is most nearly described by an idealized 1s#u2#d2s#u1#d2p#u3#d electron configuration.] #IA.3.3 Natural Bond Orbital Analysis#N #N0 The next segments of the output summarize the results of NBO analysis. The first segment reports on details of the search for an NBO natural Lewis structure: NATURAL BOND ORBITAL ANALYSIS: Occupancies Lewis Structure Low High Occ. ------------------- ----------------- occ occ Cycle Thresh. Lewis Non-Lewis CR BD 3C LP (L) (NL) Dev ============================================================================= 1(1) 1.90 17.95048 0.04952 2 6 0 1 0 0 0.02 ---------- ------------------------------------------------------------------- Structure accepted: No low occupancy Lewis orbitals #T @seg #NNormally, there is but one cycle of the NBO search (cf. the ``RESONANCE'' keyword, Section B.6.6). The table sum- marizes a variety of information for each cycle: the occu- pancy thresh#|old for a `good' pair in the NBO search; the total populations of Lewis and non-Lewis NBOs; the number of core (CR), 2-center bond (BD), 3-center bond (3C), and lone pair (LP) NBOs in the natural Lewis structure; the number of low-occupancy Lewis (L) and `high-occupancy' (> 0.1e) non- Lewis (NL) orbitals; and the maximum deviation (`Dev') of any formal bond order from a nominal estimate (NAO Wiberg bond index) for the structure. [If the latter exceeds 0.1, additional NBO searches are initiated (indicated by the parenthesized number under `Cycle') for alternative Lewis structures.] The Lewis structure is accepted if all orbi- tals of the formal Lewis structure exceed the occupancy thresh#|old (default, 1.90 electrons). 0 #NNext follows a more detailed breakdown of the Lewis and non-Lewis occupancies into core, valence, and Rydberg shell contributions: WARNING: 1 low occupancy (<1.9990e) core orbital found on July 11, 1995 - 18 - C 1 -------------------------------------------------------- Core 3.99853 ( 99.963% of 4) Valence Lewis 13.95195 ( 99.657% of 14) ================== ============================ Total Lewis 17.95048 ( 99.725% of 18) ----------------------------------------------------- Valence non-Lewis 0.03977 ( 0.221% of 18) Rydberg non-Lewis 0.00975 ( 0.054% of 18) ================== ============================ Total non-Lewis 0.04952 ( 0.275% of 18) ----- --------------------------------------------------- #T @seg #NThis shows the general quality of the natural Lewis struc- ture description in terms of the percentage of the total electron density (e.g., in the above case, about 99.7%). The table also exhibits the relatively important role of the valence non-Lewis orbitals (i.e., the six valence antibonds, NBOs 23-28) relative to the extra-valence orbitals (the 13 Rydberg NBOs 10-22) in the slight departures from a local- ized Lewis structure model. (In this case, the table also includes a warning about a carbon core orbital with slightly less than double occupancy.) (Occupancy) Bond orbital/ Coefficients/ Hybrids ----- ----- --------------------------------------------------------------------- 1. (1.99858) BD ( 1) C 1- N 2 ( 40.07%) 0.6330* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 59.93%) 0.7742* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 2. (1.99860) BD ( 1) C 1- H 3 ( 59.71%) 0.7727* C 1 s( 25.78%)p 2.88( 74.22%) -0.0002 -0.5077 0.0069 0.1928 0.0098 0.8396 -0.0046 0.0000 0.0000 ( 40.29%) 0.6347* H 3 s(100.00%) -1.0000 -0.0030 July 11, 1995 - 19 - 3. (1.99399) BD ( 1) C 1- H 4 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 0.7070 -0.0103 ( 38.98%) 0.6243* H 4 s(100.00%) 1.0000 0.0008 4. (1.99399) BD ( 1) C 1- H 5 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 -0.7070 0.0103 ( 38.98%) 0.6243* H 5 s(100.00%) 1.0000 0.0008 5. (1.99442) BD ( 1) N 2- H 6 ( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 -0.7070 -0.0093 ( 31.88%) 0.5646* H 6 s(100.00%) 1.0000 0.0020 6. (1.99442) BD ( 1) N 2- H 7 ( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 0.7070 0.0093 ( 31.88%) 0.5646* H 7 s(100.00%) 1.0000 0.0020 7. (1.99900) CR ( 1) C 1 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0003 0.0000 -0.0002 0.0000 0.0001 0.0000 0.0000 0.0000 8. (1.99953) CR ( 1) N 2 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 9. (1.97795) LP ( 1) N 2 s( 17.85%)p 4.60( 82.15%) 0.0000 0.4225 0.0002 0.2360 -0.0027 0.8749 -0.0162 0.0000 0.0000 July 11, 1995 - 20 - 10. (0.00105) RY*( 1) C 1 s( 1.57%)p62.84( 98.43%) 0.0000 -0.0095 0.1248 -0.0305 0.7302 -0.0046 0.6710 0.0000 0.0000 11. (0.00034) RY*( 2) C 1 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0146 0.9999 12. (0.00022) RY*( 3) C 1 s( 56.51%)p 0.77( 43.49%) 0.0000 -0.0023 0.7517 -0.0237 0.3710 -0.0094 -0.5447 0.0000 0.0000 13. (0.00002) RY*( 4) C 1 s( 41.87%)p 1.39( 58.13%) 14. (0.00116) RY*( 1) N 2 s( 1.50%)p65.53( 98.50%) 0.0000 -0.0062 0.1224 0.0063 0.0371 0.0197 0.9915 0.0000 0.0000 15. (0.00044) RY*( 2) N 2 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0132 0.9999 16. (0.00038) RY*( 3) N 2 s( 33.38%)p 2.00( 66.62%) 0.0000 0.0133 0.5776 0.0087 -0.8150 -0.0121 -0.0405 0.0000 0.0000 17. (0.00002) RY*( 4) N 2 s( 65.14%)p 0.54( 34.86%) 18. (0.00178) RY*( 1) H 3 s(100.00%) -0.0030 1.0000 19. (0.00096) RY*( 1) H 4 s(100.00%) -0.0008 1.0000 20. (0.00096) RY*( 1) H 5 s(100.00%) -0.0008 1.0000 21. (0.00122) RY*( 1) H 6 s(100.00%) -0.0020 1.0000 22. (0.00122) RY*( 1) H 7 s(100.00%) -0.0020 1.0000 23. (0.00016) BD*( 1) C 1- N 2 ( 59.93%) 0.7742* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 July 11, 1995 - 21 - -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 40.07%) -0.6330* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 24. (0.01569) BD*( 1) C 1- H 3 ( 40.29%) 0.6347* C 1 s( 25.78%)p 2.88( 74.22%) 0.0002 0.5077 -0.0069 -0.1928 -0.0098 -0.8396 0.0046 0.0000 0.0000 ( 59.71%) -0.7727* H 3 s(100.00%) 1.0000 0.0030 25. (0.00769) BD*( 1) C 1- H 4 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 -0.7070 0.0103 ( 61.02%) -0.7812* H 4 s(100.00%) -1.0000 -0.0008 26. (0.00769) BD*( 1) C 1- H 5 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 0.7070 -0.0103 ( 61.02%) -0.7812* H 5 s(100.00%) -1.0000 -0.0008 27. (0.00426) BD*( 1) N 2- H 6 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 0.7070 0.0093 ( 68.12%) -0.8253* H 6 s(100.00%) -1.0000 -0.0020 28. (0.00426) BD*( 1) N 2- H 7 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 -0.7070 -0.0093 ( 68.12%) -0.8253* H 7 s(100.00%) July 11, 1995 - 22 - -1.0000 -0.0020 #NNext follows the main listing of NBOs, displaying the form and occupancy of the complete set of NBOs that span the input AO space: #T @seg #NFor each NBO (1-28), the first line of printout shows the occupancy (between 0 and 2.0000 electrons) and unique label of the NBO. This label gives the type (``BD'' for 2-center bond, ``CR'' for 1-center core pair, ``LP'' for 1-center valence lone pair, ``RY*'' for 1-center Rydberg, and ``BD*'' for 2-center antibond, the unstarred and starred labels corresponding to Lewis and non-Lewis NBOs, respectively), a serial number (1, 2,... if there is a single, double,... bond between the pair of atoms), and the atom(s) to which the NBO is affixed. [For example, the first NBO in the sam- ple output is the 2-center bond (with 1.99858 electrons) between carbon (atom 1) and nitrogen (atom 2), the gma #dCN#u bond.] The next lines summarize the natural atomic hybrids #Ih#N#dA#u of which the NBO is composed, giving the percentage (100|#Ic#N#dA#u|#u2#d) of the NBO on each hybrid (in parentheses), the polarization coefficient #Ic#N#dA#u, the atom label, and a hybrid label showing the #Isp#N#u #d composition (percentage #Is#N-character, #Ip#N-character, etc.) of each #Ih#N#dA#u. [For example, the gma #dCN#u NBO is formed from an #Isp#N#u3.61#d hybrid (78.3% #Ip#N- character) on carbon interacting with an #Isp#N#u2.24#d hybrid (69.1% #Ip#N-character) on nitrogen, gma #dCN#u = 0.633(#Isp#N#u3.61#d)#dC#u + 0.774(#Isp#N#u2.24#d)#dN#u corresponding roughly to the qualitative concept of interacting #Isp#N#u3#d hybrids (75% #Ip#N-character) and the higher electronegativity (larger polarization coeffi- cient) of N.] Below each NHO label is the set of coeffi- cients that specify how the NHO is written explicitly as a linear combination of NAOs on the atom. The order of NAO coefficients follows the numbering of the NAO tables. [For example, in the first NBO entry, the carbon hybrid #Ih#N#dC#u of the gma #dCN#u bond has largest coefficients for the 2#und#d and 4#uth#d NAOs, corresponding to the approximate description #Ih#N#dC#u ~= minus 0.4653(2#Is#N)#dC#u minus 0.8808(2#Ip#N#dx#u)#dC#u in terms of the valence NAOs of the carbon atom.] In the CH#d3#uNH#d2#u example, the NBO search finds the C-N bond July 11, 1995 - 23 - (NBO 1), three C-H bonds (NBOs 2, 3, 4), two N-H bonds (NBOs 5, 6), N lone pair (NBO 9), and C and N core pairs (NBOs 7, 8) of the expected Lewis structure. NBOs 10-28 represent the residual non-Lewis NBOs of low occupancy. In this example, it is also interesting to note the slight asymmetry of the three gma #dCH#u NBOs, and the slightly higher occupancy (0.01569 #Ivs.#N 0.0077 electrons) in the gma *#<#dC#d1#uH#d3#u#u antibond (NBO 24) lying #Itrans#N to the nitrogen lone pair. #IA.3.4 NHO Directional Analysis#N 0 The next segment of output summarizes the angular proper- ties of the natural hybrid orbitals: NHO Directionality and "Bond Bending" (deviations from line of nuclear centers) [Thresholds for printing: angular deviation > 1.0 degree] hybrid p-character > 25.0% orbital occupancy > 0.10e Line of Centers Hybrid 1 Hybrid 2 --------------- ------------------- ------------------ NBO Theta Phi Theta Phi Dev Theta Phi Dev =============================================================================== 1. BD ( 1) C 1- N 2 90.0 5.4 -- -- -- 90.0 182.4 3.0 3. BD ( 1) C 1- H 4 35.3 130.7 34.9 129.0 1.0 -- -- -- 4. BD ( 1) C 1- H 5 144.7 130.7 145.1 129.0 1.0 -- -- -- 5. BD ( 1) N 2- H 6 144.7 310.7 145.0 318.3 4.4 -- -- -- 6. BD ( 1) N 2- H 7 35.3 310.7 35.0 318.3 4.4 -- -- -- 9. LP ( 1) N 2 -- -- 90.0 74.8 -- -- -- -- #T @seg #NThe `direction' of a hybrid is specified in terms of the polar (heta ) and azimuthal (hi ) angles (in the ESS coordinate system) of the vector describing its #Ip#N-component. The hybrid direction is compared with the direction of the line of centers between the two nuclei to determine the `bending' of the bond, expressed as the deviation angle (``Dev,'' in degrees) July 11, 1995 - 24 - between these two directions. For example, in the CH#d3#uNH#d2#u case shown above, the nitrogen NHO of the gma #dCN#u bond (NBO 1) is bent away from the line of C-N centers by 3.09egree , whereas the carbon NHO is approximately aligned with the C-N axis (within the 1.09egree threshold for print- ing). The N-H bonds (NBOs 5, 6) are bent even further (4.49egree ). The information in this table is often useful in anticipating the direction of geometry changes resulting from geometry optimization (viz., likely reduced pyramidali- zation of the -NH#d2#u group to relieve the nitrogen bond `kinks' found in the tetrahedral Pople-Gordon geometry). #IA.3.5 Perturbation Theory Energy Analysis#N 0 The next segment summarizes the second-order perturbative estimates of `donor-acceptor' (bond-antibond) interactions in the NBO basis: Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis Threshold for printing: 0.50 kcal/mol E(2) E(j)-E(i) F(i,j) Donor NBO (i) Acceptor NBO (j) kcal/mol a.u. a.u. =============================================================================== within unit 1 2. BD ( 1) C 1- H 3 / 14. RY*( 1) N 2 0.84 2.18 0.038 3. BD ( 1) C 1- H 4 / 26. BD*( 1) C 1- H 5 0.52 1.39 0.024 3. BD ( 1) C 1- H 4 / 27. BD*( 1) N 2- H 6 3.03 1.37 0.057 4. BD ( 1) C 1- H 5 / 25. BD*( 1) C 1- H 4 0.52 1.39 0.024 4. BD ( 1) C 1- H 5 / 28. BD*( 1) N 2- H 7 3.03 1.37 0.057 5. BD ( 1) N 2- H 6 / 10. RY*( 1) C 1 0.56 1.78 0.028 5. BD ( 1) N 2- H 6 / 25. BD*( 1) C 1- H 4 2.85 1.51 0.059 6. BD ( 1) N 2- H 7 / 10. RY*( 1) C 1 0.56 1.78 0.028 6. BD ( 1) N 2- H 7 / 26. BD*( 1) C 1- H 5 2.85 1.51 0.059 7. CR ( 1) C 1 / 16. RY*( 3) N 2 0.61 13.11 0.080 7. CR ( 1) C 1 / 18. RY*( 1) H 3 1.40 11.99 0.116 7. CR ( 1) C 1 / 19. RY*( 1) H 4 1.55 11.99 0.122 July 11, 1995 - 25 - 7. CR ( 1) C 1 / 20. RY*( 1) H 5 1.55 11.99 0.122 8. CR ( 1) N 2 / 10. RY*( 1) C 1 1.51 16.23 0.140 8. CR ( 1) N 2 / 12. RY*( 3) C 1 0.84 16.77 0.106 8. CR ( 1) N 2 / 21. RY*( 1) H 6 0.61 16.26 0.089 8. CR ( 1) N 2 / 22. RY*( 1) H 7 0.61 16.26 0.089 9. LP ( 1) N 2 / 24. BD*( 1) C 1- H 3 8.13 1.13 0.086 9. LP ( 1) N 2 / 25. BD*( 1) C 1- H 4 1.46 1.14 0.037 9. LP ( 1) N 2 / 26. BD*( 1) C 1- H 5 1.46 1.14 0.037 #T @seg #NThis is carried out by examining all possible interactions between `filled' (donor) Lewis-type NBOs and `empty' (accep- tor) non-Lewis NBOs, and estimating their energetic impor- tance by 2nd-order perturbation theory. Since these interactions lead to loss of occupancy from the localized NBOs of the idealized Lewis structure into the empty non- Lewis orbitals (and thus, to departures from the idealized Lewis structure description), they are referred to as `delo- calization' corrections to the zeroth-order natural Lewis structure. For each donor NBO (#Ii#N) and acceptor NBO (#Ij#N), the stabilization energy E(2) associated with delo- calization (``2e-stabilization'') #Ii7 arr j#N is estimated as E(2) = Delta E#dij#u = q#di#u quo <> where #Iq#N#di#u is the donor orbital occupancy, \psilon #di#u, \psilon #dj#u are diagonal elements (orbital ener- gies) and F(i,j) is the off-diagonal NBO Fock matrix ele- ment. [In the example above, the #In#N#dN#u7 arr gma *#<#dCH#u interaction between the nitrogen lone pair (NBO 8) and the antiperiplanar C#d1#u-H#d3#u antibond (NBO 24) is seen to give the strongest stabilization, 8.13 kcal/mol.] As the heading indicates, entries are included in this table only when the interaction energy exceeds a default threshold of 0.5 kcal/mol. #IA.3.6 NBO Summary#N 0 Next appears a condensed summary of the principal NBOs, showing the occupancy, orbital energy, and the qualitative pattern of delocalization interactions associated with each: Natural Bond Orbitals (Summary): Princi- pal Delocalizations July 11, 1995 - 26 - NBO Occupancy Energy (geminal,vicinal,remote) =============================================================================== Molecular unit 1 (CH5N) 1. BD ( 1) C 1- N 2 1.99858 -0.89908 2. BD ( 1) C 1- H 3 1.99860 -0.69181 14(v) 3. BD ( 1) C 1- H 4 1.99399 -0.68892 27(v),26(g) 4. BD ( 1) C 1- H 5 1.99399 -0.68892 28(v),25(g) 5. BD ( 1) N 2- H 6 1.99442 -0.80951 25(v),10(v) 6. BD ( 1) N 2- H 7 1.99442 -0.80951 26(v),10(v) 7. CR ( 1) C 1 1.99900 -11.04131 19(v),20(v),18(v),16(v) 8. CR ( 1) N 2 1.99953 -15.25927 10(v),12(v),21(v),22(v) 9. LP ( 1) N 2 1.97795 -0.44592 24(v),25(v),26(v) 10. RY*( 1) C 1 0.00105 0.97105 11. RY*( 2) C 1 0.00034 1.02120 12. RY*( 3) C 1 0.00022 1.51414 13. RY*( 4) C 1 0.00002 1.42223 14. RY*( 1) N 2 0.00116 1.48790 15. RY*( 2) N 2 0.00044 1.59323 16. RY*( 3) N 2 0.00038 2.06475 17. RY*( 4) N 2 0.00002 2.25932 18. RY*( 1) H 3 0.00178 0.94860 19. RY*( 1) H 4 0.00096 0.94464 20. RY*( 1) H 5 0.00096 0.94464 21. RY*( 1) H 6 0.00122 0.99735 22. RY*( 1) H 7 0.00122 0.99735 23. BD*( 1) C 1- N 2 0.00016 0.57000 24. BD*( 1) C 1- H 3 0.01569 0.68735 25. BD*( 1) C 1- H 4 0.00769 0.69640 26. BD*( 1) C 1- H 5 0.00769 0.69640 27. BD*( 1) N 2- H 6 0.00426 0.68086 28. BD*( 1) N 2- H 7 0.00426 0.68086 ------------------------------- Total Lewis 17.95048 ( 99.7249%) Valence non-Lewis 0.03977 ( 0.2209%) Rydberg non-Lewis 0.00975 ( 0.0542%) ------------------------------- Total unit 1 18.00000 (100.0000%) Charge unit 1 0.00000 #T @seg #NThis table allows one to quickly identify the principal delocalizing acceptor orbitals associated with each donor NBO, and their topological relationship to this NBO, i.e., whether attached to the same atom (geminal, ``g''), to an July 11, 1995 - 27 - adjacent bonded atom (vicinal, ``v''), or to a more remote (``r'') site. These acceptor NBOs will generally correspond to the principal `delocalization tails' of the NLMO associated with the parent donor NBO. [For example, in the table above, the nitrogen lone pair (NBO 9) is seen to be the lowest- occupancy (1.97795 electrons) and highest-energy (minus 0.44592 a.u.) Lewis NBO, and to be primarily delocalized into antibonds 24, 25, 26 (the vicinal gma *#<#dCH#u NBOs). The summary at the bottom of the table shows that the Lewis NBOs 1-9 describe about 99.7% of the total electron density, with the remaining non-Lewis density found primarily in the valence-shell antibonds (particularly, NBO 24).] #HSection B: NBO USER'S GUIDE#N #BB.1 INTRODUCTION TO THE NBO USER'S GUIDE AND NBO KEYLISTS#N 0 Section B constitutes the general user's guide to the NBO program. It assumes that the user has an installed elec- tronic structure system (ESS) with attached NBO program, a general idea of what the NBO method is about, and some acquaintance with standard NBO terminology and output data. If you are completely inexperienced in these areas, read Section A (General Introduction and Installation) for the necessary background to this Section. 0 The User's Guide describes how to use the NBO program by modifying your input file to the ESS program to get some NBO output. The modification consists of adding a list of #Ikeywords#N in a prescribed #Ikeylist#N format. Four dis- tinct keylist ($KEY) types are recognized ($NBO, $CORE, $CHOOSE, and $DEL), and these will be described in turn in Sections B.2-B.5. Some of the details of inserting NBO keylists into the input file depend on the details of your ESS method, and are described in the appropriate Appendix for the ESS. However, the general form of NBO keylists and the meaning and function of each keyword are identical for all versions (insofar as the option is meaningful for the ESS), and are described herein. 0 The four keylist types have common rules of syntax: Keyl- ist delimiters are identified by a ``$'' prefix. Each keyl- ist begins with the parent keylist name (e.g., ``$NBO''), followed by any number of keywords, and ended with the word ``$END''; for example, #T July 11, 1995 - 28 - $NBO keyword1 keyword2 . . . $END !comment #N(The allowed keyword entries for each type of keylist are described in Sections B.2-B.5.) The keylist is ``free for- mat,'' with keywords separated by commas or any number of spaces. An NBO option is activated by simply including its keyword in the appropriate keylist. The order of keywords in the principal $NBO keylist does not matter, but multiple keylists must be given in the order (1) $NBO, (2) $CORE, (3) $CHOOSE, (4) $DEL of presentation in Sections B.2-B.5. Key- words may be typed in upper or lower case, and will be echoed near the top of the NBO output. A $KEY list can be continued to any number of lines, but all the entries of a $KEY list must appear in a distinct set of lines, starting with the $KEY name on the first line and ending with the closing $END on the last line (i.e., no two $KEY lists should share parts of the same line). As the above example indicates, any line in the keylist input may terminate with an exclamation point (!) followed by `comment' of your choice; the ``!'' is considered to terminate the line, and the trailing `comment' is ignored by the program. #BB.2 THE $NBO KEYLIST#N #IB.2.1 Overview of $NBO keywords#N 0 The $NBO keylist is the principal means of specifying NBO job options and controlling output, and must precede any other keylist ($CORE, $CHOOSE, or $DEL) in your input file. The allowed keywords that can appear in a $NBO keylist are grouped as follows: #IJob Control Keywords:#N #IJob Threshold Keywords:#N #IMa- trix Output Keywords:#N #IOther Output Control Keywords:#N #IPrint Level Control:#N PRINT=n Keywords are first listed and described according to these formal groupings in Sections B.2.2-B.2.6. Section B.6 illustrates the effect of commonly used $NBO keywords (as well as other $KEY lists) on the successive stages of NAO/NBO/NLMO transformation and subsequent energy or dipole analysis, with sample output for these keyword options. 0 Some keywords of the $NBO keylist require (or allow) numerical values or other parameters to specify their exact function. In this case, the numerical value or parameter must immediately follow the keyword after an equal sign (=) or any number of blank spaces. Examples: #T E2PERT=2.5 LFNPR 16 NBOMO=W25 #N(The equal sign is recommended, and will be used in the remaining examples.) July 11, 1995 - 29 - [0 Although the general user's interaction with the NBO pro- grams is usually through the documented keywords of Sections B.2.2-B.2.6, some additional `semi-documented' keywords are listed in Section B.2.7 which may be of interest to the spe- cialist.] #IB.2.2 Job Control Keywords#N 0 The keywords in this group activate or deactivate basic tasks to be performed by the NBO programs, or change the way the NBO search is conducted. Each keyword is described in terms of the option it activates (together with an indica- tion of where the option is useful): #IOPTION DESCRIPTION#N Request Natural Population Analysis and printing of NPA sum- mary tables (Section A.3.2). This keyword also activates calculation of NAOs, except for semi-empirical ESS methods. Request calculation of NBOs and printing of the main NBO table (Section A.3.3). Request printing of the NBO summary table (Section A.3.6). This combines elements of the NBO table and 2nd-order per- turbation theory analysis table (see below) in a convenient form for recognizing the principal delocalization patterns. Request search for highly delocalized structures (Section B.6.6). The NBO search normally aborts when one or more Lewis NBOs has less than the default occupancy threshold of 1.90 electrons for a `good' electron pair. When the RESO- NANCE keyword is activated, this threshold is successively lowered in 0.10 decrements to 1.50, and the NBO search repeated to find the best Lewis structure within each occu- pancy threshold. The program returns with the best overall Lewis structure (lowest total non-Lewis occupancy) found in these searches. (Useful for benzene and other highly delo- calized molecules.) Request that no bonds (2-center NBOs) are to be formed in the NBO procedure (Section B.6.7). The resulting NBOs will then simply be 1-center atomic hybrids. (Useful for highly ionic species.) Request search for 3-center bonds (Section B.6.8). The nor- mal default is to search for only 1- and 2-center NBOs. (Useful for diborane and other electron-deficient `bridged' species.) Skip the computation of NBOs, i.e., only determine NAOs and perform natural population analysis. (Useful when only NPA is desired.) Compute and print out the summary table of Natural Localized Molecular Orbitals (Section B.6.2). NLMOs are similar to July 11, 1995 - 30 - Boys or Edmiston-Ruedenberg LMOs, but more efficiently calcu- lated. (Useful for `semi-localized' description of an SCF or correlated wavefunction.) Activated automatically by all keywords that pertain to NLMOs (e.g., AONLMO, SPNLMO, DIPOLE). Note that the SKIPBO keyword has higher precedence than other keywords in this list, so that keywords with which it is implicitly in conflict (e.g., NBO, 3CBOND, NLMO) will be ignored if SKIPBO is included in the $NBO keylist. #IB.2.3 Job Threshold Keywords#N 0 The keywords in this group also activate new tasks to be performed by the NBO program, but these keywords may be modified by one or more parameters (thresholds) that control the precise action to be taken. (In each case the keywords may also be used without parameters, accepting the default values [in brackets].) #IOPTION DESCRIPTION#N Request the NHO Directional Analysis table (Section A.3.4). The three parameters [and default values] have the following significance: = threshold angular deviation for printing = threshold percentage #Ip#N-character for printing = threshold NBO occupancy for printing #NParameter values may be separated by a space or a comma. Example:#T BEND=2,10,1.9 #NThis example specifies that the bond-bending table should only include entries for angular deviations of at least 29egree (ang), hybrids of at least 10% #Ip#N-character (pct), and NBOs of occupancy at least 1.9 electrons (occ). Request the Perturbation Theory Energy Analysis table (Sec- tion A.3.5), where = threshold energy (in kcal/mol) for printing Entries will be printed for NBO donor-acceptor interaction energies that exceed the `eval' threshold. Example:#T E2PERT=5.0 #NThis example would print only interactions of at least 5 kcal/mol (i.e., only the single entry for the 8.13 kcal/mol #In#N#dN#u7 arr gma *#<#dCH#u interaction in the output of Section A.3.5). July 11, 1995 - 31 - Request the Molecular Dipole Moment Analysis table (Section B.6.3), where = threshold dipole moment (Debye) for printing The program will carry out a decomposition of the total molecular dipole moment in terms of localized NLMO and NBO contributions, including all terms whose contribution (in vector norm) exceeds the `dval' threshold. Example:#T DIPOLE=0.1 #NThis example would print out dipole contributions of all NBOs (and their delocalization interactions) of magnitude ge 0.1hsp D. #NBoth the BEND and E2PERT keywords are activated by default at the standard PRINT level option (see Section B.2.6); to get an example of dipole moment analysis, include the keyl- ist #T $NBO DIPOLE $END #Nin your input file. Note that the DIPOLE keyword leads to an analysis in terms of both NBOs and NLMOs, so that the NLMO keyword (Section B.2.2) is automatically activated in this case. #IB.2.4 Matrix Output Keywords#N 0 The keywords in this group activate the printing of vari- ous matrices to the output file, or their writing to (or reading from) external disk files. The large number of key- words in this group provide great flexibility in printing out the details of the successive transformations, eps1 eps2 or the matrices of various operators in the natural local- ized basis sets. This ordered sequence of transformations forms the basis for naming the keywords. #_Keyword Names#/ 0 The keyword for printing the matrix for a particular basis transformation, IN7 arr OUT, is constructed from the corresponding acronymns for the two sets in the generic form ``INOUT''. For example, the transformation AO7 arr NBO is keyed as ``AONBO'', while that from NBOs to NLMOs is correspondingly ``NBONLMO''. The transformations are always specified in the ordered sequence shown above (i.e., ``AONBO'' is allowed, but ``NBOAO'' is an unrecognized `backward' keyword). Keywords are recognized for #Iall possible#N transformations from the input AOs to other sets July 11, 1995 - 32 - (NAO, NHO, NBO, NLMO, MO, or the pre-orthogonal PNAO, PNHO, PNBO, PNLMO sets) in the overall sequence leading to canonical MOs, i.e., AONAO AONHO AONBO AONLMO AOMO AOPNAO AOPNHO AOPNBO AOPNLMO and from each of the orthonormal natural localized sets to sets lying to the right in the sequence, i.e., NAONHO NAONBO NAONLMO NAOMO NHONBO NHONLMO NHOMO NBONLMO NBOMO NLMOMO The matrix T#dIN,OUT#u for a specified IN7 arr OUT transform has rows labelled by the IN set and columns labelled by the OUT set. 0 One can also print out the matrix representations of the Fock matrix (F), density matrix (DM), or dipole moment matrix (DI) in the input AO set or any of the natural local- ized sets (NAO, NHO, NBO, or NLMO). The corresponding key- word is constructed by combining the abbreviation (M) for the operator with that for the set (SET) in the generic form ``MSET''. For example, to print the Fock matrix (F) in the NBO set, use the keyword ``FNBO'', or to print the dipole matrix in the NLMO basis, use ``DINLMO''. (For the dipole matrix keywords, all three vector components will be printed.) One can also print out elements of the overlap matrix (S) in the input AO basis or any of the `pre- orthogonal' sets (PNAO, PNHO, PNBO, or PNLMO), using, e.g., ``SPNAO'' for the overlap matrix in the PNAO basis. The complete set of allowed keywords for operator matrices is: FAO FNAO FNHO FNBO FNLMO DMAO DMNAO DMNHO DMNBO DMNLMO DIAO DINAO DINHO DINBO DINLMO SAO SPNAO SPNHO SPNBO SPNLMO Other desired transformations can be readily obtained from the keyword transformations by matrix multiplication. #_Keyword Parameters#/ 0 Each generic matrix keyword (``MATKEY'') can include a July 11, 1995 - 33 - parameter that specifies the output operation to be performed on the matrix. The allowed MATKEY parameters are of two types (three for AONAO, NAONBO; see below): (print out the matrix in the standard output file, 'c' columns) (write out the matrix to disk file #In#N) #NThe first (P[c]) parameter is used to control output to the standard output file. When the MATKEY keyword is inserted in the $NBO keylist with no parameters, the matrix is by default printed (in its entirety) in the standard out- put file. Thus, ``MATKEY=P'' would be equivalent to ``MAT- KEY'', with no parameters. The complete `P[c]' form of the print parameter serves to truncate the printed matrix output to a specified number of columns [c]. For example, to print out only the first 16 columns of a matrix, use the form #T MATKEY=P16 (print 16 columns) #NFor certain matrices, one can also restrict printing to only the valence (VAL) or Lewis (LEW) columns with modified `[c]' specifiers. For the transformations to MOs, use the form #T MATKEY=PVAL (print core + valence MO columns only) #Nwhere ``MATKEY'' is AOMO, NAOMO, NHOMO, NBOMO, or NLMOMO (only). This will print out only the occupied MOs and the lowest few unoccupied MOs, e.g., the six lowest virtual MOs of the methylamine example (Section A.3), though not neces- sarily those with pre#|dominant valence character. Simi- larly, for the transformations to NBOs or NLMOs, use the form #T MATKEY=PLEW (print Lewis orbital columns only) #Nwhere ``MATKEY'' is AONBO, NHONBO, NAONBO, AONLMO, NAONLMO, NHONLMO, NBONLMO (or AOMO, NAOMO, NHOMO, NBOMO, NLMOMO). This prints out the Lewis NBOs or occupied MOs only, e.g., only the nine occupied NBOs or MOs of the methy- lamine example. Judicious use of these print parameters keeps printed output within reasonable bounds in calcula- tions with large basis sets. #NThe second type of MATKEY parameter (W[n]) is used to write the matrix (in its entirety) to a specified disk file [n]. By default, each keyword transformation matrix is associated with a particular logical file number (LFN) in the range 31-49, as shown in the table below: July 11, 1995 - 34 - #NWhen the ``MATKEY=Wn'' keyword is inserted in the $NBO keylist with no `n' specifier, the matrix is by default written out (in its entirety) to this LFN. Thus, ``MATKEY=W'' is equivalent to ``MATKEY=Wn'' if ``n'' is the default LFN for that keyword. Use the ``Wn'' parameter to direct output to any non-default LFN disk file. For exam- ple, the keyword #T AONBO=W22 #Nwould write out the AO7 arr NBO transformation to LFN = 22 (rather than the default LFN = 37). 0 The format of the printed output under the print `P' parameter differs from that written to an external file under the `W' parameter. The `P' output (intended for a human reader) includes an identifying label for each row, and gives the numerical entries to somewhat lesser precision (F8.4 format) than the corresponding `W' output (F15.9 for- mat), which is usually intended as input to another program. Use the ``MATKEY=W6'' keyword to route the more precise `W' form of the matrix to the standard output file, LFN 6. 0 For the AONAO, NAONBO matrices (only), one can also include a read parameter (R), #T AONAO=Rn NAONBO=Rn #Nwhich causes the matrix to be input to the program from LFN #In#N. This parameter has the effect of `freezing' orbitals to a set prescribed in the input file (thus bypass- ing the NBO optimization of these orbitals for the molecular system). For example, the keyword ``NAONBO=R44'' would have the effect of freezing the NAO7 arr NBO transformation coef- ficients to the form specified in LFN 44 (perhaps written with the ``NAONBO=W44'' keyword in a previous calculation on isolated molecules, and now to be used in a calculation on a molecular complex). Similarly, the keyword ``AONAO=R45'' could be used to force the analysis of an excited state to be carried out in terms of the NAOs of the ground state (previously written out with the ``AONAO=W45'' keyword). #IB.2.5 Other Output Control Keywords#N 0 The keywords in this group also help to control the I/O produced by a specified set of job options, and thus supple- ment the keywords of the previous section. However, the keywords of this section `steer' the flow of information that is routinely produced by the NBO program (or can be passed through from the ESS program) without materially affecting the actual jobs performed by the NBO program. The options associated with each keyword are tabulated below: #IOPTION DESCRIPTION#N July 11, 1995 - 35 - Set the logical file number (LFN) for NBO program output. The default LFN is #In#N = 6, the usual LFN for output from the ESS program. This option can be used to steer the NBO section of the job output to a desired file. Example:#T LFNPR=25 (re-direct NBO output to LFN 25) #N Request additional details of the NBO search. This option (primarily for programming and debugging purposes) records details of the NBO loops over atoms and atom pairs, enroute to the final NBOs. Request print-out of the NAO-Wiberg Bond Index array and related valency indices (Section B.6.5). The elements of this array are the sums of squares of off-diagonal density matrix elements between pairs of atoms in the NAO basis, and are the NAO counterpart of the Wiberg bond index [K. Wiberg, Tetrahedron #B24#N, 1083-1096 (1968)]. (This bond index is routinely used to `screen' atom pairs for possible bonding in the NBO search, but the values are not printed unless the BNDIDX keyword is activated.) Request writing of information concerning the AO basis set (geometrical positions, orbital exponents, contraction coef- ficients, etc.) to an external file, LFN 31. This is a por- tion of the information needed by the ORB#|PLOT orbital con- tour plotting programs (cf. ``PLOT'' keyword below.) _1. _R_e_q_u_e_s_t _w_r_i_t_i_n_g _o_f #_I_a_l_l#_N _f_i_l_e_s _r_e_q_u_i_r_e_d _b_y _o_r_b_i_t_a_l _c_o_n_t_o_u_r _p_l_o_t_t_i_n_g _p_r_o_g_r_a_m_s _O_R_B#|_P_L_O_T. _T_h_i_s _a_c_t_i_v_a_t_e_s _t_h_e _A_O_I_N_F_O _k_e_y_w_o_r_d, _a_s _w_e_l_l _a_s _a_l_l _t_h_e _n_e_c_e_s_s_a_r_y _m_a_t_r_i_x _o_u_t_p_u_t _k_e_y_w_o_r_d_s (_A_O_N_B_O=_W_3_7, _e_t_c.) _t_h_a_t _c_o_u_l_d _b_e _r_e_q_u_i_r_e_d _f_o_r _O_R_B_P_L_O_T. _R_e_q_u_e_s_t _w_r_i_t_i_n_g _t_h_e _F_I_L_E_4_7 `_a_r_c_h_i_v_e' _f_i_l_e _t_o _e_x_t_e_r_n_a_l _d_i_s_k _f_i_l_e _L_F_N = #_I_n#_N (_o_r, _i_f ``=_n'' _i_s _n_o_t _p_r_e_s_e_n_t, _t_o _t_h_e _d_e_f_a_u_l_t _L_F_N = _4_7). _T_h_i_s _f_i_l_e _c_a_n _s_e_r_v_e _a_s _t_h_e _i_n_p_u_t _f_i_l_e _t_o _r_u_n _t_h_e _G_E_N_N_B_O _p_r_o_g_r_a_m _i_n _s_t_a_n_d-_a_l_o_n_e _m_o_d_e, _t_o _r_e_p_e_a_t _t_h_e _N_B_O _a_n_a_l_y_s_i_s (_p_o_s_s_i_b_l_y _w_i_t_h _n_e_w _j_o_b _o_p_t_i_o_n_s) _w_i_t_h_o_u_t _r_e_p_e_a_t_- _i_n_g _t_h_e _c_a_l_c_u_l_a_t_i_o_n _o_f _t_h_e _w_a_v_e_f_u_n_c_t_i_o_n (_S_e_c_t_i_o_n _B._7). _R_e_q_u_e_s_t _w_r_i_t_i_n_g _t_h_e _N_B_O _d_i_r_e_c_t _a_c_c_e_s_s _f_i_l_e (_D_A_F) _t_o _e_x_t_e_r_n_a_l _d_i_s_k _f_i_l_e _L_F_N = #_I_n#_N (_o_r, _i_f ``=_n'' _i_s _n_o_t _p_r_e_s_e_n_t, _t_o _t_h_e _d_e_f_a_u_l_t _L_F_N =_4_8). #_I_B._2._6 _P_r_i_n_t _L_e_v_e_l _K_e_y_w_o_r_d_s#_N _0 _T_h_e _k_e_y_w_o_r_d ``_P_R_I_N_T=_n'' (#_I_n#_N = _0-_4) _c_a_n _b_e _u_s_e_d _t_o _g_i_v_e _c_o_n_v_e_n_i_e_n_t, _f_l_e_x_i_b_l_e _c_o_n_t_r_o_l _o_f _a_l_l _N_B_O _o_u_t_p_u_t _i_n _t_e_r_m_s _o_f _a _s_p_e_c_i_f_i_e_d _p_r_i_n_t _l_e_v_e_l #_I_n#_N. _T_h_i_s _k_e_y_w_o_r_d _a_c_t_i_v_a_t_e_s _g_r_o_u_p_s _o_f _k_e_y_w_o_r_d_s _i_n _a _h_e_i_r_a_r_c_h_i_c_a_l _m_a_n_n_e_r, _a_n_d _t_h_u_s _i_n_c_r_e_m_e_n_t_a_l_l_y _i_n_c_r_e_a_s_e_s _t_h_e _v_o_l_u_m_e _o_f _o_u_t_p_u_t, _r_a_n_g_i_n_g _f_r_o_m #_I_n_o#_N _N_B_O _o_u_t_- _p_u_t (_P_R_I_N_T=_0) _t_o _a _c_o_n_s_i_d_e_r_a_b_l_e _v_o_l_u_m_e _o_f _d_e_t_a_i_l (_P_R_I_N_T=_4). _T_h_e _k_e_y_w_o_r_d_s _a_s_s_o_c_i_a_t_e_d _w_i_t_h _e_a_c_h _p_r_i_n_t _l_e_v_e_l _a_r_e _t_a_b_u_l_a_t_e_d _b_e_l_o_w [_d_e_f_a_u_l_t _v_a_l_u_e, _P_R_I_N_T=_2]: July 11, 1995 - 36 - _F_o_r _e_a_c_h _p_r_i_n_t _l_e_v_e_l #_I_n#_N, _t_h_e _N_B_O _o_u_t_p_u_t _w_i_l_l _i_n_c_l_u_d_e _i_t_e_m_s _a_c_t_i_v_a_t_e_d _b_y _t_h_e _l_i_s_t_e_d _k_e_y_w_o_r_d_s, _a_s _w_e_l_l _a_s _a_l_l _i_t_e_m_s _f_r_o_m _l_o_w_e_r _p_r_i_n_t _l_e_v_e_l_s. _0 _W_h_e_n _a_d_d_i_t_i_o_n_a_l _k_e_y_w_o_r_d_s _a_r_e _i_n_c_l_u_d_e_d _w_i_t_h _a ``_P_R_I_N_T=_n'' _k_e_y_w_o_r_d _i_n _t_h_e $_N_B_O _k_e_y_l_i_s_t, _t_h_e _N_B_O _o_u_t_p_u_t _i_n_c_l_u_d_e_s _t_h_e _a_d_d_i_t_i_o_n_a_l _k_e_y_w_o_r_d _i_t_e_m_s _a_s _w_e_l_l _a_s _t_h_o_s_e _i_m_p_l_i_e_d _b_y _t_h_e _p_r_i_n_t _l_e_v_e_l. _T_h_i_s _c_a_n _b_e _u_s_e_d _t_o _t_a_i_l_o_r _t_h_e _N_B_O _o_u_t_p_u_t _t_o _v_i_r_t_u_a_l_l_y _a_n_y _s_e_l_e_c_t_i_o_n _o_f _o_u_t_p_u_t _i_t_e_m_s. _F_o_r _e_x_a_m_p_l_e, _t_h_e _k_e_y_l_i_s_t #_T $_N_B_O _P_R_I_N_T=_2 _N_L_M_O _F_N_B_O=_P _N_A_O_M_O=_P_1_1 $_E_N_D #_N_w_o_u_l_d _a_d_d _t_o _t_h_e _s_t_a_n_d_a_r_d _m_e_t_h_y_l_a_m_i_n_e _o_u_t_p_u_t _f_i_l_e _o_f _S_e_c_- _t_i_o_n _A._3 _a_n _N_L_M_O _s_u_m_m_a_r_y _t_a_b_l_e, _t_h_e _F_o_c_k _m_a_t_r_i_x _i_n _t_h_e _N_B_O _b_a_s_i_s, _a_n_d _t_h_e _t_r_a_n_s_f_o_r_m_a_t_i_o_n _c_o_e_f_f_i_c_i_e_n_t_s _f_o_r _t_h_e _f_i_r_s_t _1_1 _m_o_l_e_c_u_l_a_r _o_r_b_i_t_a_l_s _i_n _t_e_r_m_s _o_f _N_A_O_s. _S_i_m_i_l_a_r_l_y, _t_o _p_r_o_d_u_c_e _t_h_e _N_P_A _l_i_s_t_i_n_g _o_n_l_y, _o_n_e _c_o_u_l_d _u_s_e #_T $_N_B_O _P_R_I_N_T=_1 _S_K_I_P_B_O $_E_N_D #_N_o_r #_T $_N_B_O _P_R_I_N_T=_0 _N_P_A $_E_N_D #_N[_T_h_e_r_e _i_s _a_c_t_u_a_l_l_y _a _s_l_i_g_h_t _d_i_f_f_e_r_e_n_c_e _b_e_t_w_e_e_n _t_h_e _t_w_o _e_x_a_m_p_l_e_s: _T_h_e _N_B_O_s _a_r_e _d_e_t_e_r_m_i_n_e_d _b_y _d_e_f_a_u_l_t (_o_n_c_e _t_h_e $_N_B_O _k_e_y_l_i_s_t _i_s _e_n_c_o_u_n_t_e_r_e_d), _e_v_e_n _i_f _a_l_l _o_u_t_p_u_t _i_s _s_u_p_p_r_e_s_s_e_d _w_i_t_h _P_R_I_N_T=_0; _i_n _t_h_e _f_i_r_s_t _e_x_a_m_p_l_e, _t_h_e _k_e_y_w_o_r_d _S_K_I_P_B_O _b_y_p_a_s_s_e_s _N_B_O _d_e_t_e_r_m_i_n_a_t_i_o_n, _w_h_e_r_e_a_s _i_n _t_h_e _s_e_c_o_n_d _e_x_a_m_p_l_e _t_h_e _N_B_O_s _a_r_e _s_t_i_l_l _d_e_t_e_r_m_i_n_e_d `_i_n _b_a_c_k_g_r_o_u_n_d.'] #_I_B._2._7 _S_e_m_i-_D_o_c_u_m_e_n_t_e_d _A_d_d_i_t_i_o_n_a_l _K_e_y_w_o_r_d_s#_N _0 _S_o_m_e _a_d_d_i_t_i_o_n_a_l _k_e_y_w_o_r_d_s _a_r_e _l_i_s_t_e_d _b_e_l_o_w _t_h_a_t _m_a_y _o_f _u_s_e _t_o _s_p_e_c_i_a_l_i_s_t_s _o_r _p_r_o_g_r_a_m _d_e_v_e_l_o_p_e_r_s: #_I_O_P_T_I_O_N _D_E_S_C_R_I_P_T_I_O_N#_N _S_e_t _t_h_e _t_h_r_e_s_h_o_l_d _o_f _o_r_b_i_t_a_l _o_c_c_u_p_a_n_c_y _d_e_s_i_r_e_d _f_o_r _b_o_n_d _o_r_b_i_t_a_l _s_e_l_e_c_t_i_o_n. _I_f _t_h_i_s _i_s _n_o_t _i_n_c_l_u_d_e_d, _t_h_e _d_e_f_a_u_l_t _o_c_c_u_p_a_n_c_y [_1._9_0] _w_i_l_l _b_e _u_s_e_d (_o_r _v_a_l_u_e_s _d_e_c_r_e_a_s_i_n_g _f_r_o_m _1._9_0 _t_o _1._5_0 _b_y _0._1_0 _s_t_e_p_s, _i_f _t_h_e _R_E_S_O_N_A_N_C_E _k_e_y_w_o_r_d _i_s _i_n_c_l_u_d_e_d). _S_e_t _t_h_e _p_r_o_j_e_c_t_i_o_n _t_h_r_e_s_h_o_l_d [_d_e_f_a_u_l_t _0._2_0] _t_o _d_e_t_e_r_m_i_n_e _i_f _a `_n_e_w' _h_y_b_r_i_d _o_r_b_i_t_a_l _h_a_s _t_o_o _h_i_g_h _o_v_e_r_l_a_p _w_i_t_h _h_y_b_r_i_d_s _p_r_e_v_i_o_u_s_l_y _f_o_u_n_d. _P_r_i_n_t _t_o_t_a_l _g_r_o_s_s _M_u_l_l_i_k_e_n _p_o_p_u_l_a_t_i_o_n_s _b_y _a_t_o_m. _P_r_i_n_t _g_r_o_s_s _M_u_l_l_i_k_e_n _p_o_p_u_l_a_t_i_o_n_s, _b_y _o_r_b_i_t_a_l _a_n_d _a_t_o_m. _R_e_v_i_s_e_s _P_A_O _t_o _P_N_A_O _t_r_a_n_s_f_o_r_m_a_t_i_o_n _m_a_t_r_i_x _b_y _p_o_s_t- _m_u_l_t_i_p_l_y_i_n_g _b_y #_B_T#_N#_d_R_y_d#_u _a_n_d #_B_T#_N#_d_r_e_d#_u [_s_e_e _t_h_e _N_P_A July 11, 1995 - 37 - _p_a_p_e_r: _A. _E. _R_e_e_d, _R. _B. _W_e_i_n_s_t_o_c_k, _a_n_d _F. _W_e_i_n_h_o_l_d, _J. _C_h_e_m. _P_h_y_s. #_B_8_3#_N, _7_3_5-_7_4_6 (_1_9_8_5)]. _I_n_p_u_t _o_r _o_u_t_p_u_t _o_f _p_u_r_e _A_O (_P_A_O) _t_o _p_r_e-_N_A_O (_P_N_A_O) _t_r_a_n_s_f_o_r_- _m_a_t_i_o_n. _T_h_e _P_A_O_s _a_r_e _A_O_s _o_f _p_u_r_e _a_n_g_u_l_a_r _m_o_m_e_n_t_u_m _s_y_m_m_e_t_r_y (_r_a_t_h_e_r _t_h_a_n _c_a_r_t_e_s_i_a_n _g_a_u_s_s_i_a_n_s). _T_h_i_s _k_e_y_w_o_r_d _c_a_n _b_e _u_s_e_d _w_i_t_h _r_e_a_d (`_R'), _w_r_i_t_e (`_W', _d_e_f_a_u_l_t _L_F_N _4_3) _o_r _p_r_i_n_t (`_P') _p_a_r_a_m_e_t_e_r_s. _P_r_i_n_t _o_u_t _t_h_e _b_o_n_d-_o_r_d_e_r _m_a_t_r_i_x (_F_o_c_k-_D_i_r_a_c _d_e_n_s_i_t_y _m_a_t_r_i_x) _i_n _t_h_e _b_a_s_i_s _s_e_t _o_f _i_n_p_u_t _A_O_s. _T_h_i_s _k_e_y_w_o_r_d _c_a_n _b_e _u_s_e_d _w_i_t_h _w_r_i_t_e (`_W', _d_e_f_a_u_l_t _L_F_N _4_9) _o_r _p_r_i_n_t (`_P') _p_a_r_a_m_e_t_e_r_s. #_B_B._3 _T_H_E $_C_O_R_E _L_I_S_T#_N _0 _I_n _t_h_e _L_e_w_i_s _s_t_r_u_c_t_u_r_e _p_i_c_t_u_r_e, _t_h_e _i_n_n_e_r `_c_o_r_e' _e_l_e_c_t_r_o_n _p_a_i_r_s _a_r_e _p_i_c_t_u_r_e_d _a_s _o_c_c_u_p_y_i_n_g _o_r_b_i_t_a_l_s _h_a_v_i_n_g _e_s_s_e_n_t_i_a_l_l_y _i_s_o_l_a_t_e_d _a_t_o_m_i_c _o_r_b_i_t_a_l _c_h_a_r_a_c_t_e_r. _I_n _N_B_O _p_a_r_l_a_n_c_e, _t_h_e_s_e _c_o_r_e _o_r_b_i_t_a_l_s _c_o_r_r_e_s_p_o_n_d _t_o _1-_c_e_n_t_e_r _u_n_h_y_b_r_i_d_i_z_e_d _N_A_O_s _o_f _n_e_a_r-_m_a_x_i_m_u_m _o_c_c_u_p_a_n_c_y, _w_h_i_c_h _a_r_e _i_s_o_l_a_t_e_d _o_n _e_a_c_h _c_e_n_t_e_r _b_e_f_o_r_e _t_h_e _m_a_i_n _N_B_O _s_e_a_r_c_h _b_e_g_i_n_s _f_o_r _l_o_c_a_l_i_z_e_d _v_a_l_e_n_c_e _e_l_e_c_t_r_o_n _p_a_i_r_s. _A _w_a_r_n_i_n_g _m_e_s_s_a_g_e _i_s _p_r_i_n_t_e_d _i_f _t_h_e _o_c_c_u_- _p_a_n_c_y _o_f _a _p_r_e_s_u_m_e_d _c_l_o_s_e_d-_s_h_e_l_l _c_o_r_e _N_B_O _f_a_l_l_s _b_e_l_o_w _1._9_9_9_0 _e_l_e_c_t_r_o_n_s (_o_r _0._9_9_9_0 _i_n _t_h_e _o_p_e_n-_s_h_e_l_l _c_a_s_e), _i_n_d_i_c_a_t_i_v_e _o_f _a _p_o_s_s_i_b_l_e _c_o_r_e-_v_a_l_e_n_c_e _m_i_x_i_n_g _e_f_f_e_c_t _o_f _p_h_y_s_i_c_a_l _s_i_g_n_i_f_i_- _c_a_n_c_e. _0 [_I_n _p_r_e_v_i_o_u_s _v_e_r_s_i_o_n_s _o_f _t_h_e _N_B_O _p_r_o_g_r_a_m, _c_o_r_e _o_r_b_i_t_a_l_s _h_a_v_i_n_g _t_h_e _e_x_p_e_c_t_e_d _p_u_r_e _a_t_o_m_i_c _c_h_a_r_a_c_t_e_r _a_r_e _f_o_u_n_d _i_n _e_s_s_e_n_t_i_a_l_l_y _a_l_l _c_a_s_e_s, _e_x_c_e_p_t _w_h_e_r_e _a_n `_a_c_c_i_d_e_n_t_a_l' _d_e_g_e_n_- _e_r_a_c_y _i_n _o_c_c_u_p_a_n_c_y _o_f _c_o_r_e _a_n_d _v_a_l_e_n_c_e _l_o_n_e _p_a_i_r_s _l_e_a_d_s _t_o _u_n_d_e_s_i_r_a_b_l_e _c_o_r_e-_v_a_l_e_n_c_e _m_i_x_i_n_g; _t_h_e _p_r_e_s_e_n_t _v_e_r_s_i_o_n _e_x_p_l_i_- _c_i_t_l_y _i_s_o_l_a_t_e_s _c_o_r_e _p_a_i_r_s _a_s _u_n_h_y_b_r_i_d_i_z_e_d _N_A_O_s _p_r_i_o_r _t_o _t_h_e _m_a_i_n _N_B_O _s_e_a_r_c_h _t_o _p_r_e_v_e_n_t _t_h_i_s _u_n_p_h_y_s_i_c_a_l _e_f_f_e_c_t.] _0 _T_h_e _N_B_O _p_r_o_g_r_a_m _c_o_n_t_a_i_n_s _a _t_a_b_l_e _g_i_v_i_n_g _t_h_e _n_o_m_i_n_a_l _n_u_m_b_e_r _o_f _c_o_r_e _o_r_b_i_t_a_l_s _t_o _b_e _i_s_o_l_a_t_e_d _o_n _e_a_c_h _t_y_p_e _o_f _a_t_o_m (_e._g., _1#_I_s#_N _f_o_r _f_i_r_s_t-_r_o_w _a_t_o_m_s _L_i-_N_e, _1#_I_s#_N, _2#_I_s#_N, _2#_I_p#_N _f_o_r _s_e_c_o_n_d-_r_o_w _a_t_o_m_s _N_a-_A_r, _e_t_c.). _A_t _t_i_m_e_s, _h_o_w_e_v_e_r, _i_t _i_s _i_n_t_e_r_e_s_t_i_n_g _t_o _e_x_a_m_i_n_e _t_h_e _e_f_f_e_c_t _o_f _a_l_l_o_w_i_n_g _c_o_r_e _o_r_b_i_t_a_l_s _t_o _m_i_x _i_n_t_o _t_h_e _b_o_n_d_i_n_g _h_y_b_r_i_d_s, _o_r _t_o _h_y_b_r_i_d_i_z_e (_p_o_l_a_r_i_z_e) _a_m_o_n_g _t_h_e_m_s_e_l_v_e_s. _T_h_i_s _c_a_n _b_e _a_c_c_o_m_p_l_i_s_h_e_d _b_y _i_n_c_l_u_d_i_n_g _a $_C_O_R_E _k_e_y_l_i_s_t _t_o _s_p_e_c_i_f_y _t_h_e _n_u_m_b_e_r _o_f _c_o_r_e _o_r_b_i_t_a_l_s _t_o _b_e _i_s_o_l_a_t_e_d _o_n _e_a_c_h _a_t_o_m_i_c _c_e_n_t_e_r, _t_h_u_s _m_o_d_i_f_y_i_n_g _t_h_e _n_o_m_i_n_a_l _c_o_r_e _t_a_b_l_e. _U_n_l_i_k_e _o_t_h_e_r _N_B_O _k_e_y_l_i_s_t_s, _t_h_e $_C_O_R_E _l_i_s_t _i_n_c_l_u_d_e_s _o_n_l_y _i_n_t_e_g_e_r_s (_r_a_t_h_e_r _t_h_a_n _k_e_y_w_o_r_d_s) _t_o _s_p_e_c_i_f_y _t_h_e _c_o_r_e _m_o_d_i_f_i_c_a_t_i_o_n_s, _b_u_t _t_h_e _r_u_l_e_s _a_r_e _o_t_h_e_r_w_i_s_e _s_i_m_i_l_a_r _t_o _t_h_o_s_e _f_o_r _o_t_h_e_r _k_e_y_l_i_s_t_s. _T_h_e $_C_O_R_E _l_i_s_t (_i_f _i_n_c_l_u_d_e_d) _m_u_s_t _f_o_l_l_o_w _t_h_e $_N_B_O _k_e_y_l_i_s_t _a_n_d _p_r_e_c_e_d_e _t_h_e $_C_H_O_O_S_E _o_r $_D_E_L _k_e_y_l_i_s_t_s. _0 _T_h_e _f_o_r_m_a_t _o_f _t_h_e $_C_O_R_E _m_o_d_i_f_i_c_a_t_i_o_n _l_i_s_t _i_s: July 11, 1995 - 38 - _T_h_e _k_e_y_w_o_r_d ``$_C_O_R_E'' _P_a_i_r_s _o_f _i_n_t_e_g_e_r_s, _o_n_e _p_a_i_r _f_o_r _e_a_c_h _c_e_n_t_e_r. _T_h_e _f_i_r_s_t _i_n_t_e_g_e_r _i_n_d_i_c_a_t_e_s _t_h_e _a_t_o_m_i_c _c_e_n_t_e_r (_i_n _t_h_e _n_u_m_b_e_r_i_n_g _o_f _t_h_e _m_a_i_n _E_S_S) _a_n_d _t_h_e _s_e_c_o_n_d _i_s _t_h_e _n_u_m_b_e_r _o_f _c_o_r_e _o_r_b_i_t_a_l_s _t_o _b_e _i_s_o_l_a_t_e_d _o_n _t_h_a_t _a_t_o_m. _N_o_t_e _t_h_a_t _a_t_o_m_i_c _c_e_n_t_e_r_s _n_o_t _i_n_c_l_u_d_e_d _i_n _t_h_e _C_O_R_E _l_i_s_t _a_r_e _a_s_s_i_g_n_e_d _d_e_f_a_u_l_t _c_o_r_e_s. _T_h_e _k_e_y_w_o_r_d ``$_E_N_D'', _t_o _i_n_d_i_c_a_t_e _t_h_e _e_n_d _o_f _c_o_r_e _i_n_p_u_t. _T_h_e _e_n_t_i_r_e _l_i_s_t _m_a_y _a_l_s_o _b_e _c_o_n_d_e_n_s_e_d _t_o _a _s_i_n_g_l_e _l_i_n_e, _b_u_t _t_h_e _w_o_r_d ``$_C_O_R_E'' _m_u_s_t _o_c_c_u_r _a_s _t_h_e _f_i_r_s_t _w_o_r_d _o_f _t_h_e _l_i_n_e _a_n_d ``$_E_N_D'' _a_s _t_h_e _l_a_s_t _w_o_r_d; _t_h_a_t _i_s, _t_h_e _c_o_r_e _m_o_d_i_f_i_c_a_- _t_i_o_n _k_e_y_l_i_s_t _c_a_n_n_o_t _c_o_n_t_i_n_u_e _o_n _a _l_i_n_e _t_h_a_t _c_o_n_t_a_i_n_s _o_t_h_e_r _k_e_y_l_i_s_t _i_n_f_o_r_m_a_t_i_o_n. _0 _T_h_e _c_o_r_e _o_r_b_i_t_a_l_s _a_r_e _i_s_o_l_a_t_e_d _b_y _o_c_c_u_p_a_n_c_y, _t_h_e _m_o_s_t _o_c_c_u_p_i_e_d _N_A_O_s _b_e_i_n_g _f_i_r_s_t _s_e_l_e_c_t_e_d, _a_n_d _f_u_l_l _s_u_b_s_h_e_l_l_s _a_r_e _i_s_o_l_a_t_e_d _a_t _a _t_i_m_e. _T_h_u_s, _f_o_r _e_x_a_m_p_l_e, _t_o _s_e_l_e_c_t _t_h_e _f_i_v_e _o_r_b_i_t_a_l_s _o_f _t_h_e #_I_n#_N = _1 _a_n_d #_I_n#_N = _2 _s_h_e_l_l_s _a_s _c_o_r_e _o_r_b_i_- _t_a_l_s, _i_t _w_o_u_l_d _m_a_k_e _n_o _d_i_f_f_e_r_e_n_c_e _t_o _s_e_l_e_c_t ``_3'' _o_r ``_4'' (_i_n_s_t_e_a_d _o_f ``_5''), _s_i_n_c_e _a_l_l _t_h_r_e_e _o_f _t_h_e_s_e _c_h_o_i_c_e_s _w_o_u_l_d _s_p_e_c_i_f_y _a _c_o_r_e _c_o_n_t_a_i_n_i_n_g _a _1#_I_s#_N, _2#_I_s#_N, _a_n_d _a_l_l _t_h_r_e_e _2#_I_p#_N _o_r_b_i_t_a_l_s. _T_h_e $_C_O_R_E _m_o_d_i_f_i_c_a_t_i_o_n _l_i_s_t _i_s _r_e_a_d _o_n_l_y _o_n_c_e, _a_n_d _a_p_p_l_i_e_s _t_o _b_o_t_h _l_p_h_a _a_n_d7777777777777 t99a999999 s99p99i99n9999 m99a99n99i99f99o99l99d99s9999 i99n9999 a99n9999 o99_p7777777777777_n-_s_h_e_l_l _c_a_l_c_u_l_a_- _t_i_o_n. _A_n _e_x_a_m_p_l_e, _a_p_p_r_o_p_r_i_a_t_e _f_o_r _N_i(_1)-_C(_2)-_O(_3) _w_i_t_h _t_h_e _i_n_d_i_- _c_a_t_e_d _n_u_m_b_e_r_i_n_g _o_f _a_t_o_m_s, _i_s _s_h_o_w_n _b_e_l_o_w: #_T $_C_O_R_E _1 _5 $_E_N_D #_N_T_h_i_s _w_o_u_l_d _d_i_r_e_c_t _t_h_e _N_B_O _p_r_o_g_r_a_m _t_o _i_s_o_l_a_t_e _o_n_l_y _5 _c_o_r_e _o_r_b_i_t_a_l_s _o_n _N_i_c_k_e_l (_a_t_o_m _1), _r_a_t_h_e_r _t_h_a_n _t_h_e _n_o_m_i_n_a_l _9 _c_o_r_e _o_r_b_i_t_a_l_s. _I_n _o_t_h_e_r _w_o_r_d_s, _o_n_l_y _1#_I_s#_N, _2#_I_s#_N, _a_n_d _2#_I_p#_N _o_r_b_i_t_a_l_s _w_i_l_l _b_e _c_o_n_s_i_d_e_r_e_d _a_s _c_o_r_e _o_r_b_i_t_a_l_s _i_n _t_h_e _s_e_a_r_c_h _f_o_r _N_B_O_s _o_f _N_i_C_O, _a_l_l_o_w_i_n_g _t_h_e _3#_I_s#_N _a_n_d _3#_I_p#_N _o_r_b_i_t_a_l_s _t_o _m_i_x _w_i_t_h _v_a_l_e_n_c_e _N_A_O_s _i_n _b_o_n_d _f_o_r_m_a_t_i_o_n. _S_i_n_c_e _t_h_e _c_a_r_b_o_n _a_n_d _o_x_y_g_e_n _a_t_o_m_s _w_e_r_e _n_o_t _i_n_c_l_u_d_e_d _i_n _t_h_e _m_o_d_i_f_i_c_a_t_i_o_n _l_i_s_t, _t_h_e _n_o_m_i_n_a_l _s_e_t _o_f _c_o_r_e _o_r_b_i_t_a_l_s (_1#_I_s#_N _o_n_l_y) _i_s _i_s_o_l_a_t_e_d _o_n _e_a_c_h _o_f _t_h_e_s_e _a_t_o_m_s. [_T_h_e _a_l_t_e_r_n_a_t_i_v_e _e_x_a_m_p_l_e #_T $_C_O_R_E _1 _0 _2 _0 _3 _0 $_E_N_D #_N(_n_o _c_o_r_e_s) _w_o_u_l_d _a_l_l_o_w _a_l_l _N_A_O_s _t_o _b_e _i_n_c_l_u_d_e_d _i_n _t_h_e _N_B_O _s_e_a_r_c_h; _t_h_i_s _w_o_u_l_d _b_e _e_q_u_i_v_a_l_e_n_t _t_o _t_h_e _d_e_f_a_u_l_t _t_r_e_a_t_m_e_n_t _i_n _t_h_e _e_a_r_l_i_e_r _v_e_r_s_i_o_n _o_f _t_h_e _p_r_o_g_r_a_m (_s_e_e _S_e_c_t_i_o_n _A._1._5).] #_B_B._4 _T_H_E $_C_H_O_O_S_E _K_E_Y_L_I_S_T (_D_I_R_E_C_T_E_D _N_B_O _S_E_A_R_C_H)#_N _0 _A $_C_H_O_O_S_E _k_e_y_l_i_s_t _r_e_q_u_e_s_t_s _t_h_a_t _t_h_e _N_B_O _s_e_a_r_c_h _b_e _d_i_r_e_c_t_e_d July 11, 1995 - 39 - _t_o _f_i_n_d _a _p_a_r_t_i_c_u_l_a_r _L_e_w_i_s _s_t_r_u_c_t_u_r_e (`_r_e_s_o_n_a_n_c_e _s_t_r_u_c_t_u_r_e') _c_h_o_s_e_n _b_y _t_h_e _u_s_e_r. (_T_h_i_s _i_s _u_s_e_f_u_l _f_o_r _t_e_s_t_i_n_g _t_h_e _a_c_c_u_- _r_a_c_y _o_f _a_l_t_e_r_n_a_t_i_v_e _r_e_s_o_n_a_n_c_e _s_t_r_u_c_t_u_r_e _r_e_p_r_e_s_e_n_t_a_t_i_o_n_s _o_f _t_h_e _w_a_v_e_f_u_n_c_t_i_o_n, _r_e_l_a_t_i_v_e _t_o _t_h_e _o_p_t_i_m_a_l _L_e_w_i_s _s_t_r_u_c_t_u_r_e _r_e_t_u_r_n_e_d _i_n _a _f_r_e_e _N_B_O _s_e_a_r_c_h.) _I_n _t_h_e $_C_H_O_O_S_E _l_i_s_t, _a _r_e_s_o_n_a_n_c_e _s_t_r_u_c_t_u_r_e _i_s _s_p_e_c_i_f_i_e_d _b_y _i_n_d_i_c_a_t_i_n_g _w_h_e_r_e _l_o_n_e _p_a_i_r_s _a_n_d _b_o_n_d_s (_i_n_c_l_u_d_i_n_g _m_u_l_t_i_p_l_e _b_o_n_d_s) _a_r_e _t_o _b_e _f_o_u_n_d _i_n _t_h_e _m_o_l_e_c_u_l_e. _I_n _s_o_m_e _c_a_s_e_s, _t_h_e _u_s_e_r _m_a_y _w_i_s_h _t_o _s_p_e_c_i_f_y _o_n_l_y _t_h_e _l_o_c_a_t_i_o_n _o_f _b_o_n_d_s, _l_e_t_t_i_n_g _t_h_e _N_B_O _a_l_g_o_- _r_i_t_h_m _s_e_e_k _t_h_e _b_e_s_t _l_o_c_a_t_i_o_n _f_o_r _l_o_n_e _p_a_i_r_s, _b_u_t _i_t _i_s _u_s_u_- _a_l_l_y _s_a_f_e_s_t _t_o _c_o_m_p_l_e_t_e_l_y _s_p_e_c_i_f_y _t_h_e _r_e_s_o_n_a_n_c_e _s_t_r_u_c_t_u_r_e, _b_o_t_h _l_o_n_e _p_a_i_r_s _a_n_d _b_o_n_d_s. _0 _T_h_e _f_o_r_m_a_t _o_f _t_h_e $_C_H_O_O_S_E _l_i_s_t _i_s: _T_h_e _k_e_y_w_o_r_d ``$_C_H_O_O_S_E'' _T_h_e _k_e_y_w_o_r_d ``_A_L_P_H_A'' (_o_n_l_y _f_o_r _o_p_e_n-_s_h_e_l_l _w_a_v_e_f_u_n_c_t_i_o_n) _I_f _o_n_e-_c_e_n_t_e_r (`_l_o_n_e') _N_B_O_s _a_r_e _t_o _b_e _s_e_a_r_c_h_e_d _f_o_r, _t_y_p_e _t_h_e _k_e_y_w_o_r_d ``_L_O_N_E'' _f_o_l_l_o_w_e_d _b_y _a _l_i_s_t _o_f _p_a_i_r_s _o_f _n_u_m_b_e_r_s, _t_h_e _f_i_r_s_t _n_u_m_b_e_r _o_f _e_a_c_h _p_a_i_r _b_e_i_n_g _t_h_e _a_t_o_m_i_c _c_e_n_t_e_r _a_n_d _t_h_e _s_e_c_o_n_d _t_h_e _n_u_m_b_e_r _o_f _v_a_l_e_n_c_e _l_o_n_e _p_a_i_r_s _o_n _t_h_a_t _a_t_o_m. _T_e_r_- _m_i_n_a_t_e _t_h_e _l_i_s_t _w_i_t_h ``_E_N_D''. (_N_o_t_e _t_h_a_t _o_n_l_y _t_h_e _o_c_c_u_p_i_e_d #_I_v_a_l_e_n_c_e#_N _l_o_n_e _p_a_i_r_s _s_h_o_u_l_d _b_e _e_n_t_e_r_e_d, _s_i_n_c_e _t_h_e _n_u_m_b_e_r _o_f _c_o_r_e _o_r_b_i_t_a_l_s _o_n _e_a_c_h _c_e_n_t_e_r _i_s _p_r_e_s_u_m_e_d _k_n_o_w_n.) _I_f _t_w_o-_c_e_n_t_e_r (`_b_o_n_d') _N_B_O_s _a_r_e _t_o _b_e _s_e_a_r_c_h_e_d _f_o_r, _t_y_p_e _t_h_e _k_e_y_w_o_r_d ``_B_O_N_D'', _f_o_l_l_o_w_e_d _b_y _t_h_e _l_i_s_t _o_f _b_o_n_d _s_p_e_c_i_f_i_e_r_s, _a_n_d _t_e_r_m_i_n_a_t_e_d _b_y ``_E_N_D''. _E_a_c_h _b_o_n_d _s_p_e_c_i_f_i_e_r _i_s _o_n_e _o_f _t_h_e _l_e_t_t_e_r_s _s_i_n_g_l_e _b_o_n_d _d_o_u_b_l_e _b_o_n_d _t_r_i_p_l_e _b_o_n_d _q_u_a_d_r_u_p_l_e _b_o_n_d _f_o_l_l_o_w_e_d _b_y _t_h_e _t_w_o _a_t_o_m_i_c _c_e_n_t_e_r_s _o_f _t_h_e _b_o_n_d (_e._g., ``_D _9 _1_6'' _f_o_r _a _d_o_u_b_l_e _b_o_n_d _b_e_t_w_e_e_n _a_t_o_m_s _9 _a_n_d _1_6). _I_f _t_h_r_e_e-_c_e_n_t_e_r _N_B_O_s _a_r_e _t_o _b_e _s_e_a_r_c_h_e_d _f_o_r, _t_y_p_e _t_h_e _k_e_y_- _w_o_r_d ``_3_C_B_O_N_D'', _f_o_l_l_o_w_e_d _b_y _t_h_e _l_i_s_t _o_f _3-_c _b_o_n_d _s_p_e_c_i_f_- _i_e_r_s, _a_n_d _t_e_r_m_i_n_a_t_e_d _b_y ``_E_N_D''. _E_a_c_h _3-_c _b_o_n_d _s_p_e_c_i_f_i_e_r _i_s _a_g_a_i_n _o_n_e _o_f _t_h_e _l_e_t_t_e_r_s ``_S'' (_s_i_n_g_l_e), ``_D'' (_d_o_u_b_l_e), ``_T'' (_t_r_i_p_l_e), _o_r ``_Q'' (_q_u_a_d_r_u_p_l_e), _f_o_l_l_o_w_e_d _b_y _t_h_r_e_e _i_n_t_e_g_e_r_s _f_o_r _t_h_e _t_h_r_e_e _a_t_o_m_i_c _c_e_n_t_e_r_s (_e._g., ``_S _4 _8 _1_0'' _f_o_r _a _s_i_n_g_l_e _t_h_r_e_e-_c_e_n_t_e_r _b_o_n_d _4-_8-_1_0). (_N_o_t_e _t_h_a_t _t_h_e _3_C_B_O_N_D _k_e_y_w_o_r_d _o_f _t_h_e $_N_B_O _k_e_y_l_i_s_t _i_s _i_m_p_l_i_c_i_t_l_y _a_c_t_i_v_a_t_e_d _i_f _3-_c _b_o_n_d_s _a_r_e _i_n_c_l_u_d_e_d _i_n _a $_C_H_O_O_S_E _l_i_s_t.) _T_h_e _w_o_r_d ``_E_N_D'' _t_o _s_i_g_n_a_l _t_h_e _e_n_d _o_f _t_h_e _l_p_h_a _s_p_i_n _l_i_s_t. _T_h_e _k_e_y_w_o_r_d ``_B_E_T_A'' (_f_o_r _o_p_e_n-_s_h_e_l_l _w_a_v_e_f_u_n_c_t_i_o_n_s) _T_h_e _i_n_p_u_t _f_o_r777777 t99a999999 s99p99i99n99,9999 s99a99_m777777 _f_o_r_m_a_t _a_s _a_b_o_v_e. _T_h_e _o_v_e_r_a_l_l $_C_H_O_O_S_E _l_i_s_t _s_h_o_u_l_d _a_l_w_a_y_s _e_n_d _w_i_t_h _t_h_e ``$_E_N_D'' _k_e_y_w_o_r_d. July 11, 1995 - 40 - _T_w_o _e_x_a_m_p_l_e_s _w_i_l_l _s_e_r_v_e _t_o _i_l_l_u_s_t_r_a_t_e _t_h_e $_C_H_O_O_S_E _f_o_r_m_a_t (_e_a_c_h _i_s _r_a_t_h_e_r _a_r_t_i_f_i_c_i_a_l, _i_n_a_s_m_u_c_h _a_s _t_h_e _s_p_e_c_i_f_i_e_d $_C_H_O_O_S_E _s_t_r_u_c_t_u_r_e _c_o_r_r_e_s_p_o_n_d_s _t_o _t_h_e `_n_o_r_m_a_l' _s_t_r_u_c_t_u_r_e _t_h_a_t _w_o_u_l_d _b_e _f_o_u_n_d _b_y _t_h_e _N_B_O _p_r_o_g_r_a_m): _T_h_e _c_l_o_s_e_d-_s_h_e_l_l _H-_b_o_n_d_e_d _c_o_m_p_l_e_x _F_H9_o_t_s _C_O, _w_i_t_h _a_t_o_m _n_u_m_b_e_r_i_n_g _F(_1)-_H(_2)9_o_t_s _C(_3)-_O(_4), _m_i_g_h_t _b_e _s_p_e_c_i_f_i_e_d _a_s #_T $_C_H_O_O_S_E _L_O_N_E _1 _3 _3 _1 _4 _1 _E_N_D _B_O_N_D _S _1 _2 _T _3 _4 _E_N_D $_E_N_D #_N_T_h_i_s _w_o_u_l_d _d_i_r_e_c_t _t_h_e _N_B_O _p_r_o_g_r_a_m _t_o _s_e_a_r_c_h _f_o_r _t_h_r_e_e _l_o_n_e _p_a_i_r_s _o_n _a_t_o_m _F(_1), _o_n_e _l_o_n_e _p_a_i_r _o_n _a_t_o_m _C(_3), _o_n_e _l_o_n_e _p_a_i_r _o_n _a_t_o_m _O(_4), _o_n_e _b_o_n_d _b_e_t_w_e_e_n _F(_1)-_H(_2), _a_n_d _t_h_r_e_e _b_o_n_d_s _b_e_t_w_e_e_n _C(_3)-_O(_4). _T_h_e _o_p_e_n-_s_h_e_l_l _F_H9_o_t_s _O#_d_2#_u _c_o_m_p_l_e_x, _w_i_t_h _a_t_o_m _n_u_m_b_e_r_i_n_g _F(_1)-_H(_2)#+...#-_O(_3)-_O(_4), _a_n_d _w_i_t_h _t_h_e _u_n_p_a_i_r_e_d _e_l_e_c_t_r_o_n_s _o_n _O#_d_2#_u _b_e_i_n_g _o_f _l_p_h_a _s_p_i_n, _m_i_g_h_t _b_e _s_p_e_c_i_f_i_e_d _a_s #_T $_C_H_O_O_S_E _A_L_P_H_A _L_O_N_E _1 _3 _3 _3 _4 _3 _E_N_D _B_O_N_D _S _1 _2 _S _3 _4 _E_N_D _E_N_D _B_E_T_A _L_O_N_E _1 _3 _3 _1 _4 _1 _E_N_D _B_O_N_D _S _1 _2 _T _3 _4 _E_N_D _E_N_D $_E_N_D #_N_N_o_t_e _t_h_a_t _t_h_i_s _e_x_a_m_p_l_e _i_n_c_o_r_p_o_r_a_t_e_s _t_h_e _i_d_e_a _o_f ``_d_i_f_- _f_e_r_e_n_t _L_e_w_i_s _s_t_r_u_c_t_u_r_e_s _f_o_r _d_i_f_f_e_r_e_n_t _s_p_i_n_s,'' _w_i_t_h _a _d_i_s_- _t_i_n_c_t _p_a_t_t_e_r_n _o_f _l_o_c_a_l_i_z_e_d _1-_c (`_l_o_n_e') _a_n_d _2-_c (`_b_o_n_d') _f_u_n_c_t_i_o_n_s _f_o_r _l_p_h_a _a_n_d7777 t99a999999 s99p99i99n99.7777 _0 _A_s _w_i_t_h _o_t_h_e_r _k_e_y_l_i_s_t_s, _t_h_e $_C_H_O_O_S_E _k_e_y_l_i_s_t _c_a_n _b_e _c_o_n_d_e_n_s_e_d _t_o _a _s_m_a_l_l_e_r _n_u_m_b_e_r _o_f _l_i_n_e_s, _a_s _l_o_n_g _a_s _n_o _l_i_n_e _i_s _s_h_a_r_e_d _w_i_t_h _a_n_o_t_h_e_r _k_e_y_l_i_s_t. _T_h_e _o_r_d_e_r _o_f _k_e_y_w_o_r_d_s _w_i_t_h_i_n _t_h_e $_C_H_O_O_S_E _k_e_y_l_i_s_t _s_h_o_u_l_d _b_e _a_s _s_h_o_w_n _a_b_o_v_e (_i._e., _A_L_P_H_A _b_e_f_o_r_e _B_E_T_A, _L_O_N_E _b_e_f_o_r_e _B_O_N_D, _e_t_c.), _b_u_t _t_h_e _o_r_d_e_r _o_f _e_n_t_r_i_e_s _w_i_t_h_i_n _a _L_O_N_E _o_r _B_O_N_D _l_i_s_t _i_s _i_m_m_a_t_e_r_i_a_l. _A $_C_O_R_E _k_e_y_l_i_s_t (_i_f _p_r_e_s_e_n_t) _m_u_s_t _p_r_e_c_e_d_e _t_h_e $_C_H_O_O_S_E _l_i_s_t. #_B_B._5 _T_H_E $_D_E_L _K_E_Y_L_I_S_T (_N_B_O _E_N_E_R_G_E_T_I_C _A_N_A_L_Y_S_I_S)#_N July 11, 1995 - 41 - #_I_B._5._1 _I_n_t_r_o_d_u_c_t_i_o_n _t_o _N_B_O _E_n_e_r_g_e_t_i_c _A_n_a_l_y_s_i_s#_N _0 _T_h_e _f_o_u_r_t_h _a_n_d _f_i_n_a_l _t_y_p_e _o_f _k_e_y_l_i_s_t _i_s _a `_d_e_l_e_t_i_o_n_s' ($_D_E_L) _k_e_y_l_i_s_t, _t_o _a_c_t_i_v_a_t_e _N_B_O _e_n_e_r_g_e_t_i_c _a_n_a_l_y_s_i_s. _T_h_i_s _a_n_a_l_y_s_i_s _i_s _p_e_r_f_o_r_m_e_d _b_y (_1) _d_e_l_e_t_i_n_g _s_p_e_c_i_f_i_e_d _e_l_e_m_e_n_t_s (_o_r _b_l_o_c_k_s _o_f _e_l_e_m_e_n_t_s) _f_r_o_m _t_h_e _N_B_O _F_o_c_k _m_a_t_r_i_x, (_2) _d_i_a_g_o_n_a_l_- _i_z_i_n_g _t_h_i_s _n_e_w _F_o_c_k _m_a_t_r_i_x _t_o _o_b_t_a_i_n _a _n_e_w _d_e_n_s_i_t_y _m_a_t_r_i_x, _a_n_d (_3) _p_a_s_s_i_n_g _t_h_i_s _d_e_n_s_i_t_y _m_a_t_r_i_x _t_o _t_h_e _S_C_F _r_o_u_t_i_n_e_s _f_o_r _a _s_i_n_g_l_e _p_a_s_s _t_h_r_o_u_g_h _t_h_e _S_C_F _e_n_e_r_g_y _e_v_a_l_u_a_t_o_r. _T_h_e _d_i_f_f_e_r_- _e_n_c_e _b_e_t_w_e_e_n _t_h_i_s `_d_e_l_e_t_i_o_n' _e_n_e_r_g_y _a_n_d _t_h_e _o_r_i_g_i_n_a_l _S_C_F _e_n_e_r_g_y _p_r_o_v_i_d_e_s _a _u_s_e_f_u_l _m_e_a_s_u_r_e _o_f _t_h_e _e_n_e_r_g_y _c_o_n_t_r_i_b_u_t_i_o_n _o_f _t_h_e _d_e_l_e_t_e_d _t_e_r_m_s. _S_i_n_c_e _a _F_o_c_k _m_a_t_r_i_x _i_s _r_e_q_u_i_r_e_d, _t_h_e _e_n_e_r_g_e_t_i_c _a_n_a_l_y_s_i_s _i_s _p_e_r_f_o_r_m_e_d _f_o_r _R_H_F _a_n_d _U_H_F _w_a_v_e_f_u_n_c_- _t_i_o_n_s _o_n_l_y. _0 _I_n_p_u_t _f_o_r _t_h_e _N_B_O _e_n_e_r_g_e_t_i_c _a_n_a_l_y_s_i_s _i_s _t_h_r_o_u_g_h _t_h_e $_D_E_L _k_e_y_l_i_s_t, _w_h_i_c_h _s_p_e_c_i_f_i_e_s _t_h_e _d_e_l_e_t_i_o_n_s _t_o _b_e _p_e_r_f_o_r_m_e_d. _M_u_l_t_i_p_l_e _a_n_a_l_y_s_e_s (_d_e_l_e_t_i_o_n_s) _c_a_n _b_e _p_e_r_f_o_r_m_e_d _d_u_r_i_n_g _a _s_i_n_- _g_l_e _j_o_b, _w_i_t_h _e_a_c_h _d_e_l_e_t_i_o_n _i_n_c_l_u_d_e_d _i_n _t_h_e _o_v_e_r_a_l_l $_D_E_L _k_e_y_l_i_s_t. _T_h_e _n_i_n_e _d_i_s_t_i_n_c_t _t_y_p_e_s _o_f _d_e_l_e_t_i_o_n_s _i_n_p_u_t _a_r_e _d_e_s_c_r_i_b_e_d _i_n _S_e_c_t_i_o_n _B._5._2 _b_e_l_o_w. _0 _T_h_e _d_e_l_e_t_i_o_n_s _k_e_y_l_i_s_t _b_e_g_i_n_s _w_i_t_h _t_h_e ``$_D_E_L'' _k_e_y_w_o_r_d. _F_o_r _t_h_e _a_n_a_l_y_s_i_s _o_f _U_H_F _w_a_v_e_f_u_n_c_t_i_o_n_s, _t_h_e _d_e_l_e_t_i_o_n_s _f_o_r _t_h_e _l_p_h_a _a_n_d777777777777 t99a999999 s99p99i99n9999 m99a99n99i99f99o99l99d99s9999 m99u99s99t9999 _b777777777777 _s_e_p_a_r_a_t_e_l_y _s_p_e_c_i_f_i_e_d (_s_e_e _S_e_c_t_i_o_n _B._5._3). _O_t_h_e_r_w_i_s_e, _t_h_e _i_n_p_u_t _f_o_r _c_l_o_s_e_d _s_h_e_l_l_s _R_H_F _a_n_d _U_H_F _i_s _i_d_e_n_t_- _i_c_a_l. _T_h_e _i_n_p_u_t _i_s _f_r_e_e _f_o_r_m_a_t _a_n_d _t_h_e _i_n_p_u_t _f_o_r _a _s_i_n_g_l_e _d_e_l_e_t_i_o_n _c_a_n _b_e _s_p_r_e_a_d _o_v_e_r _a_s _m_a_n_y _l_i_n_e_s _a_s _d_e_s_i_r_e_d. _T_h_e _d_e_s_i_r_e_d _d_e_l_e_t_i_o_n_s _s_h_o_u_l_d _b_e _l_i_s_t_e_d _o_n_e _a_f_t_e_r _t_h_e _o_t_h_e_r. _A_f_t_e_r _t_h_e _l_a_s_t _d_e_l_e_t_i_o_n, _t_h_e _w_o_r_d ``$_E_N_D'' _s_i_g_n_a_l_s _t_h_e _e_n_d _o_f _t_h_e _k_e_y_l_i_s_t. |<<_3//_1_8//_7_2//+_0//+_1_3>>|7______________________99______________________ #_B_W_A_R_N_I_N_G#_N _I_f _s_y_m_m_e_t_r_y _i_s _u_s_e_d, _o_n_e _m_u_s_t _b_e _c_a_r_e_f_u_l _t_o _o_n_l_y _d_o _d_e_l_e_- _t_i_o_n_s _t_h_a_t _w_i_l_l _p_r_e_s_e_r_v_e _t_h_e _s_y_m_m_e_t_r_y _o_f _t_h_e _e_l_e_c_t_r_o_n_i_c _w_a_v_e_f_u_n_c_t_i_o_n!! _I_f _t_h_i_s _i_s _n_o_t _d_o_n_e, _t_h_e _e_n_e_r_g_y _o_f _t_h_e _d_e_l_e_- _t_i_o_n _w_i_l_l _b_e _i_n_c_o_r_r_e_c_t _b_e_c_a_u_s_e _t_h_e _a_s_s_u_m_p_t_i_o_n _i_s _m_a_d_e _i_n _e_v_a_l_u_a_t_i_n_g _t_h_e _e_n_e_r_g_y _t_h_a_t _t_h_e _o_r_i_g_i_n_a_l _s_y_m_m_e_t_r_y _s_t_i_l_l _e_x_i_s_t_s, _a_n_d _t_h_e _v_a_r_i_a_t_i_o_n_a_l _p_r_i_n_c_i_p_l_e _m_a_y _b_e _v_i_o_l_a_t_e_d. (_F_o_r _e_x_a_m_p_l_e, _i_f _s_y_m_m_e_t_r_y _i_s _u_s_e_d _f_o_r _e_t_h_a_n_e, _i_s _i_s _p_e_r_m_i_s_s_i_b_l_e _t_o _d_o _a ``_N_O_S_T_A_R'' _d_e_l_e_t_i_o_n, _b_u_t _n_o_t _t_h_e _d_e_l_e_t_i_o_n _o_f _a _s_i_n_- _g_l_e _C-_H _a_n_t_i_b_o_n_d.) _T_h_e _r_e_m_e_d_y _i_s _n_o_t _t_o _u_s_e _s_y_m_m_e_t_r_y _i_n _t_h_e _S_C_F _c_a_l_c_u_l_a_t_i_o_n. _0 _I_n _d_e_s_c_r_i_b_i_n_g _t_h_e _d_e_l_e_t_i_o_n _t_y_p_e_s, _u_s_e _i_s _m_a_d_e _o_f _t_h_e _t_e_r_m_s ``_m_o_l_e_c_u_l_a_r _u_n_i_t'' _a_n_d ``_c_h_e_m_i_c_a_l _f_r_a_g_m_e_n_t.'' _T_h_e _N_B_O _p_r_o_- _g_r_a_m _l_o_o_k_s _a_t _t_h_e _c_h_e_m_i_c_a_l _b_o_n_d_i_n_g _p_a_t_t_e_r_n _p_r_o_d_u_c_e_d _b_y _t_h_e _b_o_n_d_i_n_g _N_B_O_s _a_n_d _i_d_e_n_t_i_f_i_e_s _t_h_e _g_r_o_u_p_s _o_f _a_t_o_m_s _t_h_a_t _a_r_e _l_i_n_k_e_d _t_o_g_e_t_h_e_r _i_n _d_i_s_t_i_n_c_t ``_m_o_l_e_c_u_l_a_r _u_n_i_t_s'' (_u_s_u_a_l_l_y July 11, 1995 - 42 - _s_y_n_o_n_y_m_o_u_s _w_i_t_h ``_m_o_l_e_c_u_l_e_s'' _i_n _t_h_e _c_h_e_m_i_c_a_l _s_e_n_s_e). _T_h_e _f_i_r_s_t _a_t_o_m _t_h_a_t _i_s _n_o_t _i_n _m_o_l_e_c_u_l_a_r _u_n_i_t _1 _w_i_l_l _b_e _i_n _m_o_l_e_c_u_l_a_r _u_n_i_t _2, _a_n_d _s_o _f_o_r_t_h. _F_o_r _e_x_a_m_p_l_e, _i_f _t_h_e _l_i_s_t _o_f _a_t_o_m_s _i_s _C(_1), _H(_2), _F(_3), _O(_4), _a_n_d _b_o_n_d_i_n_g _N_B_O_s _a_r_e _f_o_u_n_d _b_e_t_w_e_e_n _C(_1)- _O(_4) _a_n_d _H(_2)-_F(_3), _t_h_e_n _m_o_l_e_c_u_l_a_r _u_n_i_t _1 _w_i_l_l _b_e _C_O _a_n_d _m_o_l_e_c_u_l_a_r _u_n_i_t _2 _w_i_l_l _b_e _H_F. _A ``_c_h_e_m_i_c_a_l _f_r_a_g_m_e_n_t'' _i_s _t_a_k_e_n _t_o _b_e _a_n_y _s_u_b_s_e_t _o_f _t_h_e _a_t_o_m_s, _u_s_u_a_l_l_y (_b_u_t _n_o_t _n_e_c_e_s_- _s_a_r_i_l_y) _i_n _t_h_e _s_a_m_e _m_o_l_e_c_u_l_a_r _u_n_i_t, _a_n_d _u_s_u_a_l_l_y (_b_u_t _n_o_t _n_e_c_e_s_s_a_r_i_l_y) _c_o_n_n_e_c_t_e_d _b_y _b_o_n_d _N_B_O_s. _T_y_p_i_c_a_l_l_y, _a _c_h_e_m_i_c_a_l _f_r_a_g_m_e_n_t _m_i_g_h_t _b_e _s_p_e_c_i_f_i_e_d _t_o _b_e _a _s_i_n_g_l_e _a_t_o_m, _t_h_e _f_o_u_r _a_t_o_m_s _o_f _a _m_e_t_h_y_l _g_r_o_u_p, _o_r _a_n_y _o_t_h_e_r `_r_a_d_i_c_a_l' _o_f _a _m_o_l_e_c_u_- _l_a_r _u_n_i_t, _i_d_e_n_t_i_f_i_e_d _b_y _g_i_v_i_n_g _t_h_e _a_t_o_m _n_u_m_b_e_r_s _o_f _w_h_i_c_h _t_h_e _f_r_a_g_m_e_n_t _c_o_n_s_i_s_t_s. #_I_B._5._2 _T_h_e _N_i_n_e _D_e_l_e_t_i_o_n _T_y_p_e_s#_N _0 _T_h_e _k_e_y_w_o_r_d_s _a_n_d _f_o_r_m_a_t _t_o _s_p_e_c_i_f_y _e_a_c_h _o_f _t_h_e _n_i_n_e _a_l_l_o_w_e_d _d_e_l_e_t_i_o_n _t_y_p_e_s _a_r_e _d_e_s_c_r_i_b_e_d _b_e_l_o_w: #_(_1) _D_e_l_e_t_i_o_n _o_f _e_n_t_i_r_e _o_r_b_i_t_a_l_s.#/ _T_h_i_s _i_s _c_a_l_l_e_d _f_o_r _b_y _t_y_p_i_n_g ``_D_E_L_E_T_E'', _t_h_e_n _t_h_e _n_u_m_b_e_r _o_f _o_r_b_i_t_a_l_s _t_o _b_e _d_e_l_e_t_e_d, _t_h_e_n _t_h_e _k_e_y_w_o_r_d ``_O_R_B_I_T_A_L'' (_o_r ``_O_R_B_I_T_A_L_S''), _t_h_e_n _t_h_e _l_i_s_t _o_f _t_h_e _o_r_b_i_t_a_l_s _t_o _b_e _d_e_l_e_t_e_d. _E_x_a_m_p_l_e: #_T_D_E_L_E_T_E _3 _O_R_B_I_T_A_L_S _1_5 _1_8 _2_9 #_N[_S_e_e _a_l_s_o _d_e_l_e_t_i_o_n _t_y_p_e_s (_4) _a_n_d (_7) _f_o_r _d_e_l_e_t_i_n_g _s_e_t_s _o_f _o_r_b_i_t_a_l_s.] |<<_3//_1_8//_7_2//+_0//+_1_1>>|7______________________99______________________ #_B_W_A_R_N_I_N_G#_N _T_h_e ``_s_i_n_g_l_e-_p_a_s_s'' _m_e_t_h_o_d _o_f _e_v_a_l_u_a_t_i_n_g _d_e_l_e_t_i_o_n _e_n_e_r_g_i_e_s _i_s _a_p_p_r_o_p_r_i_a_t_e _o_n_l_y _f_o_r _d_e_l_e_t_i_o_n_s _o_f #_I_l_o_w#_N-_o_c_c_u_p_a_n_c_y (_n_o_n-_L_e_w_i_s) _o_r_b_i_t_a_l_s, _f_o_r _w_h_i_c_h _t_h_e _l_o_s_s _o_f _s_e_l_f-_c_o_n_s_i_s_t_e_n_c_y _i_n _t_h_e _C_o_u_l_o_m_b _a_n_d _e_x_c_h_a_n_g_e _p_o_t_e_n_t_i_a_l_s (_d_u_e _t_o _r_e_d_i_s_t_r_i_b_u_- _t_i_o_n _o_f _t_h_e _e_l_e_c_t_r_o_n _d_e_n_s_i_t_y _o_f _d_e_l_e_t_e_d _o_r_b_i_t_a_l_s) _i_s _s_m_a_l_l _c_o_m_p_a_r_e_d _t_o _t_h_e _n_e_t _e_n_e_r_g_y _c_h_a_n_g_e _o_f _d_e_l_e_t_i_o_n. _I_t _i_s _f_u_n_d_a_- _m_e_n_t_a_l_l_y _e_r_r_o_n_e_o_u_s _t_o _d_e_l_e_t_e #_I_h_i_g_h#_N-_o_c_c_u_p_a_n_c_y (_L_e_w_i_s) _o_r_b_i_t_a_l_s _b_y _t_h_i_s _p_r_o_c_e_d_u_r_e. #_(_2) _D_e_l_e_t_i_o_n _o_f _s_p_e_c_i_f_i_c _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s.#/ _T_h_i_s _i_s _c_a_l_l_e_d _f_o_r _b_y _t_y_p_i_n_g ``_D_E_L_E_T_E'', _t_h_e_n _t_h_e _n_u_m_b_e_r _o_f _e_l_e_m_e_n_t_s _t_o _b_e _d_e_l_e_t_e_d, _t_h_e_n _t_h_e _k_e_y_w_o_r_d ``_E_L_E_M_E_N_T'' (_o_r ``_E_L_E_M_E_N_T_S''), _t_h_e_n _t_h_e _l_i_s_t _o_f _t_h_e _e_l_e_m_e_n_t_s _t_o _b_e _d_e_l_e_t_e_d (_e_a_c_h _a_s _a _p_a_i_r _o_f _i_n_t_e_g_e_r_s). _E_x_a_m_p_l_e: #_T_D_E_L_E_T_E _3 _E_L_E_M_E_N_T_S _1 _1_5 _3 _1_9 _2_3 _2 July 11, 1995 - 43 - #_N_T_h_i_s _e_x_a_m_p_l_e _w_o_u_l_d _r_e_s_u_l_t _i_n _t_h_e _z_e_r_o_i_n_g _o_f _t_h_e _f_o_l_l_o_w_i_n_g _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s: (_1,_1_5), (_1_5,_1), (_3,_1_9), (_1_9,_3), (_2_3,_2), (_2,_2_3). [_S_e_e _a_l_s_o _d_e_l_e_t_i_o_n _t_y_p_e_s (_3), (_5), (_6), (_8), (_9) _f_o_r _d_e_l_e_t_i_n_g _s_e_t_s _o_f _e_l_e_m_e_n_t_s.] #_(_3) _D_e_l_e_t_i_o_n _o_f _o_f_f-_d_i_a_g_o_n_a_l _b_l_o_c_k_s _o_f _t_h_e _F_o_c_k _m_a_t_r_i_x.#/ _E_a_c_h _b_l_o_c_k _i_s _s_p_e_c_i_f_i_e_d _b_y _t_w_o _s_e_t_s _o_f _o_r_b_i_t_a_l_s, _a_n_d _a_l_l _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s _i_n _c_o_m_m_o_n _b_e_t_w_e_e_n _t_h_e_s_e _t_w_o _s_e_t_s _a_r_e _s_e_t _t_o _z_e_r_o. _T_h_i_s _i_s _c_a_l_l_e_d _f_o_r _b_y _t_y_p_i_n_g ``_Z_E_R_O'', _t_h_e_n _t_h_e _n_u_m_b_e_r _o_f _o_f_f-_d_i_a_g_o_n_a_l _b_l_o_c_k_s _t_o _b_e _z_e_r_o_e_d, _a_n_d _t_h_e_n, _f_o_r _e_a_c_h _b_l_o_c_k, _t_h_e _f_o_l_l_o_w_i_n_g: (_1) _t_h_e _d_i_m_e_n_s_i_o_n_s _o_f _t_h_e _b_l_o_c_k, _s_e_p_a_r_a_t_e_d _b_y _t_h_e _w_o_r_d ``_B_Y'' (_e._g., ``_6 _B_Y _3'' _i_f _t_h_e _f_i_r_s_t _s_e_t _h_a_s _6 _o_r_b_i_t_a_l_s _a_n_d _t_h_e _s_e_c_o_n_d _s_e_t _h_a_s _3 _o_r_b_i_t_a_l_s); (_2) _t_h_e _l_i_s_t _o_f _o_r_b_i_t_a_l_s _i_n _t_h_e _f_i_r_s_t _s_e_t; (_3) _t_h_e _l_i_s_t _o_f _o_r_b_i_t_a_l_s _i_n _t_h_e _s_e_c_o_n_d _s_e_t. _A_n _e_x_a_m_p_l_e _i_s _s_h_o_w_n _b_e_l_o_w: #_T _Z_E_R_O _2 _B_L_O_C_K_S _2 _B_Y _5 _3 _4 _9 _1_0 _1_1 _1_4 _1_9 _3 _B_Y _2 _1 _2 _7 _2_0 _2_4 #_N_T_h_i_s _w_i_l_l _s_e_t _t_h_e _f_o_l_l_o_w_i_n_g _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s _t_o _z_e_r_o: (_3,_9), (_3,_1_0), (_3,_1_1), (_3,_1_4), (_3,_1_9), (_9,_3), (_1_0,_3), (_1_1,_3), (_1_4,_3), (_1_9,_3), (_4,_9), (_4,_1_0), (_4,_1_1), (_4,_1_4), (_4,_1_9), (_9,_4), (_1_0,_4), (_1_1,_4), (_1_4,_4), (_1_9,_4), (_1,_2_0), (_1,_2_4), (_2,_2_0), (_2,_2_4), (_7,_2_0), (_7,_2_4) (_2_0,_1), (_2_4,_1), (_2_0,_2), (_2_4,_2), (_2_0,_7), (_2_4,_7) [_U_s_u_a_l_l_y, _i_n _s_t_u_d_y_i_n_g _t_h_e _t_o_t_a_l _d_e_l_o_c_a_l_i_z_a_t_i_o_n _f_r_o_m _o_n_e _m_o_l_e_c_u_l_a_r _u_n_i_t _t_o _a_n_o_t_h_e_r, _i_t _i_s _m_u_c_h _e_a_s_i_e_r _t_o _u_s_e _d_e_l_e_t_i_o_n _t_y_p_e (_8) _b_e_l_o_w. _S_i_m_i_l_a_r_l_y, _i_n _s_t_u_d_y_i_n_g _t_h_e _t_o_t_a_l _d_e_l_o_c_a_l_i_- _z_a_t_i_o_n _f_r_o_m _o_n_e _c_h_e_m_i_c_a_l _f_r_a_g_m_e_n_t _t_o _a_n_o_t_h_e_r, _i_t _i_s _e_a_s_i_e_r _t_o _u_s_e _d_e_l_e_t_i_o_n _t_y_p_e (_9).] #_(_4) _D_e_l_e_t_i_o_n _o_f _a_l_l _R_y_d_b_e_r_g _a_n_d _a_n_t_i_b_o_n_d _o_r_b_i_t_a_l_s.#/ _T_h_e _R_y_d_b_e_r_g _a_n_d _a_n_t_i_b_o_n_d _o_r_b_i_t_a_l_s _a_r_e _t_h_e _n_o_n-_L_e_w_i_s _N_B_O _o_r_b_i_t_a_l_s _t_h_a_t _h_a_v_e _s_t_a_r_s _i_n _t_h_e_i_r _l_a_b_e_l_s (_R_Y*, _B_D*) _i_n _t_h_e _N_B_O _a_n_a_l_y_s_i_s _o_u_t_p_u_t. _T_o _d_e_l_e_t_e _a_l_l _t_h_e_s_e _o_r_b_i_t_a_l_s, _s_i_m_p_l_y _e_n_t_e_r ``_N_O_S_T_A_R''. _T_h_e _r_e_s_u_l_t _o_f _t_h_i_s _d_e_l_e_t_i_o_n _i_s _t_h_e _e_n_e_r_g_y _o_f _t_h_e _i_d_e_a_l_i_z_e_d _N_B_O _n_a_t_u_r_a_l _L_e_w_i_s _s_t_r_u_c_t_u_r_e, _w_i_t_h _a_l_l _L_e_w_i_s _N_B_O_s _d_o_u_b_l_y _o_c_c_u_p_i_e_d. (_U_n_l_i_k_e _o_t_h_e_r _d_e_l_e_t_i_o_n_s, _i_n _w_h_i_c_h July 11, 1995 - 44 - _t_h_e_r_e _i_s _a _s_l_i_g_h_t _l_o_s_s _o_f _v_a_r_i_a_t_i_o_n_a_l _s_e_l_f-_c_o_n_s_i_s_t_e_n_c_y _d_u_e _t_o _t_h_e _r_e_d_i_s_t_r_i_b_u_t_e_d _o_c_c_u_p_a_n_c_y _o_f _t_h_e _d_e_l_e_t_e_d _o_r_b_i_t_a_l_s, _t_h_e _r_e_s_u_l_t _o_f _a ``_N_O_S_T_A_R'' _d_e_l_e_t_i_o_n _c_o_r_r_e_s_p_o_n_d_s _r_i_g_o_r_o_u_s_l_y _t_o _t_h_e _v_a_r_i_- _a_t_i_o_n_a_l _e_x_p_e_c_t_a_t_i_o_n _v_a_l_u_e _o_f _t_h_e _d_e_t_e_r_m_i_n_a_n_t _o_f _d_o_u_b_l_y _o_c_c_u_- _p_i_e_d _L_e_w_i_s _N_B_O_s). #_(_5) _D_e_l_e_t_i_o_n _o_f _a_l_l _v_i_c_i_n_a_l _d_e_l_o_c_a_l_i_z_a_t_i_o_n_s.#/ _T_o _d_e_l_e_t_e _a_l_l _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s _b_e_t_w_e_e_n _L_e_w_i_s _N_B_O_s _a_n_d _t_h_e _v_i_c_i_n_a_l _n_o_n-_L_e_w_i_s _N_B_O_s, _s_i_m_p_l_y _e_n_t_e_r ``_N_O_V_I_C''. #_(_6) _D_e_l_e_t_i_o_n _o_f _a_l_l _g_e_m_i_n_a_l _d_e_l_o_c_a_l_i_z_a_t_i_o_n_s.#/ _T_o _d_e_l_e_t_e _a_l_l _F_o_c_k _m_a_t_r_i_x _e_l_e_m_e_n_t_s _b_e_t_w_e_e_n _L_e_w_i_s _N_B_O_s _a_n_d _t_h_e _g_e_m_i_n_a_l _n_o_n-_L_e_w_i_s _N_B_O_s, _s_i_m_p_l_y _e_n_t_e_r ``_N_O_G_E_M''. #_(_7) _D_e_l_e_t_i_o_n _o_f _a_l_l _s_t_a_r_r_e_d (_a_n_t_i_b_o_n_d/_R_y_d_b_e_r_g) _o_r_b_i_t_a_l_s _o_n _a _p_a_r_t_i_c_u_l_a_r _m_o_l_e_c_u_l_a_r _u_n_i_t.#/ _T_h_i_s _i_s _c_a_l_l_e_d _f_o_r _b_y _t_y_p_i_n_g ``_D_E_S_T_A_R'', _t_h_e_n _t_h_e _n_u_m_b_e_r _o_f _m_o_l_e_c_u_l_a_r _u_n_i_t_s _t_o _b_e _d_e#|_s_t_a_r_r_e_d, _t_h_e_n _t_h_e _k_e_y_w_o_r_d ``_U_N_I_T'' (_o_r ``_U_N_I_T_S''), _t_h_e_n _t_h_e _l_i_s_t _o_f _u_n_i_t_s. _E_x_a_m_p_l_e: #_T_D_E_S_T_A_R _2 _U_N_I_T_S _3 _4 #_N #_(_8) _Z_e_r_o_i_n_g _a_l_l _d_e_l_o_c_a_l_i_z_a_t_i_o_n _f_r_o_m _o_n_e _m_o_l_e_c_u_l_a_r _u_n_i_t _t_o _a_n_o_t_h_e_r.#/ _T_h_i_s _i_s _c_a_l_l_e_d _f_o_r _b_y _t_y_p_i_n_g ``_Z_E_R_O'', _t_h_e_n _t_h_e _n_u_m_b_e_r _o_f _d_e_l_o_c_a_l_i_z_a_t_i_o_n_s _t_o _z_e_r_o, _t_h_e_n _t_h_e _k_e_y_w_o_r_d ``_D_E_L_O_C_A_L_I_Z_A_T_I_O_N'' (_c_a_n _b_e _a_b_b_r_e_v_i_a_t_e_d _t_o ``_D_E_L_O_C''), _a_n_d _t_h_e_n, _f_o_r _e_a_c_h _d_e_l_o_- _c_a_l_i_z_a_t_i_o_n, _t_h_e _w_o_r_d ``_F_R_O_M'', _t_h_e _n_u_m_b_e_r _o_f _t_h_e _d_o_n_o_r _u_n_i_t, _t_h_e _w_o_r_d ``_T_O'', _a_n_d _t_h_e _n_u_m_b_e_r _o_f _t_h_e _a_c_c_e_p_t_o_r _u_n_i_t. _E_x_a_m_p_l_e: #_T_Z_E_R_O _2 _D_E_L_O_C _F_R_O_M _1 _T_O _2 _F_R_O_M _2 _T_O _1 #_N_T_h_e _a_b_o_v_e _e_x_a_m_p_l_e _w_o_u_l_d _z_e_r_o #_I_a_l_l#_N _i_n_t_e_r_m_o_l_e_c_u_l_a_r _d_e_l_o_- _c_a_l_i_z_a_t_i_o_n_s _b_e_t_w_e_e_n _u_n_i_t_s _1 _a_n_d _2 (_i._e., _b_o_t_h _17 _a_r_r _2 _a_n_d _2 7_a_r_r _1). _T_h_e _e_f_f_e_c_t _i_s _t_o _r_e_m_o_v_e _a_l_l _F_o_c_k