Re: CCL:Summary of "rotating a density matrix"

Hi Thomas,
 I have not read the articles you mention, but solved the problem
 of rotating d-functions (6D set) myself around 1993.  Below you
 can find the relevant comment cut from a fortran77 program.
 Expanding to f-functions (10F set) should be analogous, probably
 including the normalization pecularity (see below).
 Kind greetings, Aldert
 A.J.A. Westra Hoekzema
 Leiden Institue of Chemistry
 The Netherlands
 *----------------------------------------------------------------------
 * ROTATE D-FUNCTIONS (LOCATED IN THE ORIGIN)
 *
 * IDENTIFY THE SET OF 6 D-FUNCTIONS WITH A TWO-DIMENSIONAL TENSOR:
 *  D(XX) = T(1,1)      D(XY) = T(1,2) + T(2,1)
 *  D(YY) = T(2,2)      D(XZ) = T(1,3) + T(3,1)
 *  D(ZZ) = T(3,3)      D(YZ) = T(2,3) + T(3,2)
 * WHERE T(I,J) INDICATES A 3X3 TENSOR WITH ELEMENT (I,J) EQUAL TO 1
 * AND THE OTHER EIGHT ELEMENTS EQUAL TO 0
 *
 * A ROTATED TENSOR GIVEN IN TERMS OF THE ORIGINAL TENSOR ELEMENTS
 * DEPENDS ON THE ROTATION-MATRIX R:
 *                3   3
 *  ROT T(I,J) = SUM SUM R(I,K) R(J,L) T(K,L)
 *               K=1 L=1
 *
 *  WITH R(K,I) = PARTIAL DERIVATIVE OF ROTATED CARTESIAN TO OLD ONE
 *
 * AFTER ROTATION A GENERAL COMBINATION OF D-FUNCTIONS PHI RESULTS IN
 * A COMBINATION EXPRESSED IN TERMS OF THE ORIGINAL COEFFICIENTS:
 *
 *  PHI  = SUM SUM C(IJ) D(IJ)
 *          I <= J
 *
 *  PHI' = ROT PHI = ROT SUM SUM C(IJ) D(IJ) = SUM SUM C(IJ) D(IJ)' =
 *                        I <= J                I <= J
 *
 *         SUM SUM C(IJ) SUM SUM R(I,K) R(J,L) T(K,L) =
 *          I <= J         K   L
 *
 *         SUM SUM SUM SUM C(IJ) { R(I,K) R(J,L) + { DELTA(K,L) - 1 }
 *          K <= L  I <= J         R(I,L) R(J,K) } D(KL)
 *
 *         WITH DELTA THE KRONECKER DELTA
 *
 * USING THE 6-D SET OF D-FUNCTIONS THERE IS ANOTHER PECULARITY. THE
 * NORMALIZATION FACTORS DIFFER BETWEEN D(XX/YY/ZZ) AND D(XY/XZ/YZ):
 *
 *  N(II) = (1/9) TO THE POWER (1/4) TIMES N(IJ) , J /= I
 *
 * THIS INTRODUCES A FACTOR IN THE ROTATED COMBINATION:
 *
 *  PHI' = SUM SUM SUM SUM C(IJ) { R(I,K) R(J,L) + { DELTA(K,L) - 1 }
 *          K <= L  I <= J         R(I,L) R(J,K) } D(KL) F(IJ,KL)
 *
 *  WITH F: II --> KK = 1  ,  II --> KL = SQRT(3)      ,
 *          IJ --> KL = 1  ,  IJ --> KK = 1 / SQRT(3)  ,  FOR EXAMPLE:
 *
 *  ROT DIJ <=> ROT ( N(IJ) IJ EXP ) = .. + N(IJ) KK EXP <=>
 *
 *  SQRT(3) / SQRT(3) N(IJ) KK EXP <=> SQRT(3) N(KK) KK EXP
 *
 * A GENERAL EXPRESSION:
 *
 *  F(IJ,KL) = 1 + DELTA(I,J) { 1 - DELTA(K,L) } { 1 / SQRT(3) - 1 }
 *
 *               + DELTA(K,L) { 1 - DELTA(I,J) } {     SQRT(3) - 1 }
 *
 * REFERENCES: -D.E. BOURNE AND P.C. KENDALL,
 *              VECTOR ANALYSIS AND CARTESIAN TENSORS, 2ND ED.,
 *              NELSON, 1980
 *             -GEORGE ARFKEN,
 *              MATHEMATICAL METHODS FOR PHYSICISTS, 2ND ED.,
 *              ACADEMIC PRESS, 1970
 *             -W.J. HEHRE, L. RADOM, P.V.R. SCHLEYER & J.A. POPLE,
 *              AB INITIO MOLECULAR ORBITAL THEORY, 1986, P19
 *----------------------------------------------------------------------
 > From chemistry-request ^at^ ccl.net  Fri Aug 23 09:10:08 2002
 > From: Thomas Exner <exner ^at^ pc.chemie.tu-darmstadt.de>
 >
 > Hi CCLers,
 >
 > I have an additional question to the rotation of the density matrix. I
 > accounted some difficulties while implementing the rotation of the d
 > orbitals as discribed by Ferre et al. (J.Comp.Chem. 23(6) 2002, 610-624)
 > and Philipp and Friesner (J.Comp.Chem. 20(14) 1999, 1468-1494). I think
 > I found one error in the equation in the second paper. But because I
 > don't really understand how the rotation matrix is affiliated, it is
 > impossible for me to figure out if there are any other errors in the
 > equation or if it is a problem with my implementation. I would
 > appriciate if someone could give me some hints how the rotation matrix
 > has to be constructed. Perhaps, this could also help me to construct a
 > rotation matrix for f orbital. Thank you very much.
 >
 > Best wishes,
 > Thomas
 >
 > --
 > ____________________________________________________________________
 >
 > Dr. Thomas Exner
 > Computational, Theoretical & Mathematical Chemistry
 > Department of Chemistry
 > University of Saskatchewan
 > 110 Science Place                                 431 Edmund Park
 > Saskatoon, SK, S7N 5C9                            Saskatoon, SK, S7H 0Z4
 > Canada                                            Canada
 >
 > phone:  +1-306-966-5357			          +1-306-382-5397
 > fax:    +1-306-966-4730
 >
 >
 > Department of Physical Chemistry
 > Physical Chemistry I
 > Darmstadt University of Technology (TUD)
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 > Germany						  Germany
 >
 > phone:  +49-6151-16 2398			  +49-6151-537165
 > fax:    +49-6151-16 4298
 > e-mail: exner ^at^ pc.chemie.tu-darmstadt.de		  texner ^at^ gmx.net
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 >
 >
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