Chem Topic: Excited States
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if I missed any important techniques.
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Dave Young
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Calculating Electronic Excited States
David Young
Division of University Computing
144 Parker Hall
Auburn University
Auburn, AL 36849
INTRODUCTION
This is an introduction to the techniques used for the calculation
of electronic excited states of molecules (sometimes called eximers).
Specifically, these are methods for obtaining wave functions for the
excited states of a molecule from which energies and other molecular properties
(such as dipole moments) can be calculated. These calculations are an
important tool for the analysis of spectroscopy, reaction mechanisms
and other excited state phenomena.
These same techniques may also be necessary to find the ground state
wave function. Determining the ground state electron configuration can
be particularly difficult for compounds with very low energy excited states.
This document is concerned only with ab initio calculations. These
are calculations in which only the theory of quantum mechanics is applied
with no experimental data used in the calculation. Some of these techniques
are also applicable to certain classes of semiempirical calculations.
This write up is not intended to be a detailed description.
It is meant to be a short introduction with a discussion of strengths and
weaknesses of various methods. This should also serve as a check list
of available methods when faced with computational difficulties. References
are provided for more detailed discussions information.
SPIN STATES
Ab initio programs attempt to compute the lowest energy state of
a specified multiplicity. Thus calculations for different spin states
will give a lowest energy state and a few of the excited states. This
is most often done to determine singlet - triplet gaps in organic molecules.
CIS
A single excitation configuration interaction (CIS) calculation is
probably the most common way to get excited state energies. This is because
it is one of the easiest calculations to perform.
A configuration interaction calculation uses molecular orbitals
that have been optimized typically with a Hartree-Fock (HF) calculation.
Generalized Valence Bond (GVB) and multi-configuration self consistent
field (MCSCF) calculations can also be used as a starting point for
a configuration interaction calculation.
A CIS calculation starts with this initial set of orbitals and
moves one electron to one of the virtual orbitals from the original
calculation. This gives a description of one of the excited states of
the molecule, but does not change the quality of the description of the
ground state as double excitation CIs do. This gives a wave function of
somewhat lesser quality than the original calculation since the orbitals
have been optimized for the ground state. Often this results in the ground
state energy being a bit low relative to the other states.
A CIS calculation is not extremely accurate. However, it has the
advantage of being able to compute many excited state energies easily.
INITIAL GUESS
If the initial guess for a calculation is very close to an
excited state wave function, the calculation may converge to that
excited state. This is typically done by doing an initial calculation
then using it's wave function with some of the orbitals switched as
the initial guess for another calculation.
The advantage of this method is that the orbitals have been
optimized for the excited state.
The disadvantage is that there is no guarantee that it will work.
If there is no energy barrier between the initial guess and the ground state
wave function, the entire calculation will converge back to the ground state.
The convergence path may take the calculation to an undesired state in
any case.
A second disadvantage of this technique applies if the state is the
same symmetry as a lower energy state. There is no guarantee that the
state obtained is completely orthogonal to the ground state. This means
that the wave function obtained may be some mix of the lower energy
state and a higher energy state. In practice, this type of calculation
only converges to a higher state if a fairly reasonable description of
the excited state wave function is obtained. Mixing tends to be a significant
concern if the orbital energies are very close together or the system is very
sensitive to correlation effects.
BLOCK DIAGONAL HAMILTONIANS
Most ab initio calculations use symmetry adapted molecular orbitals.
Under this scheme, the Hamiltonian matrix is block diagonal meaning that
every molecular orbital will behave according to the symmetry of one of the
irreducible representations of the point group. No orbitals will be described
by a mixing of different irreducible representations.
Some programs such as COLUMBUS, DMOL and GAMESS actually set up a
separate matrix for each irriducible representation and solve them separately.
Such programs give the user the option of defining how many electrons
are of each irreducible representation. This defines the symmetry of the
wave function. In this case the resulting wave function is the lowest
energy wave function of a particular symmetry.
This is a very good way to get excited states which differ in
symmetry from the ground state and are the lowest energy state within that
symmetry.
HIGHER ROOTS OF A CI
For configuration interaction calculations of double excitations
or higher, it is possible to solve the CI super-matrix for the second root,
third root, fourth root, etc.
This is a very reliable way to get a high quality wave function for
the first few excited states. For higher excited states, CPU times become
very large since more iterations are generally needed to converge the
CI calculation.
NEGLECTING A BASIS FUNCTION
Some programs, such as COLUMBUS, allow a calculation to be done
with some orbitals completely neglected from the calculation. For example,
in a transition metal compound you could work with four d functions so that
the calculation would have no way to occupy the function that was left out.
This is a reliable way to get an excited state wave function even
when it is not the lowest energy wave function of that symmetry. However
it might take a bit of work to construct the input file depending upon the
individual program.
IMPOSING ORTHOGONALITY - DFT TECHNIQUES
Traditionally, excited states have not been one of the strong
points of density functional theory. This is due to the difficulty
in ensuring orthogonality to the ground state wave function when
no wave functions are being used in the calculation.
The easiest excited states to find using density functional
theory techniques are those which are the lowest state of a given
symmetry thus using a ground state calculation.
A promising techniques is one which uses a variational bound
for the average of the first M states of a molecule.
A few other options have been examined. However, there is not
yet a large enough volume of work applying DFT to excited states to
predict the reliability of any of these techniques.
IMPOSING ORTHOGONALITY - QMC TECHNIQUES
Quantum Monte Carlo (QMC) methods are computations which use a
statistical integration to calculate integrals which could not be
evaluated analytically. These calculations can be extremely accurate,
but often at the expense of enormous CPU times.
There are a number of methods for getting excited state energies
from QMC calculations. These methods will only be mentioned here and
are explained more fully in the text by Hammond, Lester and Reynolds
referenced at the end of this document.
Computations done in imaginary time can yield an excited state
energy by a transformation of the energy decay curve.
If an accurate description of the ground state is already available,
an excited state description can be obtained by forcing the wave function
to be orthogonal to the ground state wave function.
Diffusion and Green's function QMC calculations are often done
using a fixed node approximation. Within this scheme, the nodal surfaces
used define the state that is obtained as well as ensuring an antisymmetric
wave function.
Matrix QMC procedures similar to configuration interaction
treatments have been devised in an attempt to calculate many states
concurrently. These methods are not yet well developed as evidenced
by oscillatory behavior in the excited state energies.
FURTHER INFORMATION
There are many books on the principles of quantum mechanics and every
physical chemistry text has an introductory treatment. The
work which I am listing here is a two volume set with each chapter
broken into a basic and advanced sections making it excellent for
both intermediate and advanced users.
C. Cohen-Tannoudji, B. Diu, F. Laloe "Quantum Mechanics Volumes I &
II"
Wiley-Interscience (1977)
For an introduction to quantum chemistry see
D. A. McQuarrie "Quantum Chemistry" University Science Books (1983)
A graduate level text on quantum chemistry is
I. N. Levine "Quantum Chemistry" Prentice Hall (1991)
For quantum Monte Carlo methods, order the following book using
ISBN 981-02-0322-5 because the title is listed incorrectly in
'Books in Print'.
B. L. Hammond, W. A. Lester, Jr., P. J. Reynolds "Monte Carlo Methods
in Ab Initio Quantum Chemistry" World Scientific (1994)
For density functional theory see
R. G. Parr, W. Yang "Density-Functional Theory of Atoms and Molecules"
Oxford (1989)
There is a comprehensive listing of all available molecular modeling
software and structural databanks, free or not, in appendix 2 of
"Reviews in Computational Chemistry Volume 6"
Ed. K. B. Lipkowitz and D. B. Boyd, VCH (1995)
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