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Exploiting non-Abelian symmetry in AO-driven CIS, RPA, and TDDFT calculations
Alex A. Granovsky

Laboratory of Chemical Cybernetics, M.V. Lomonosov Moscow State University, Moscow, Russia
February 15, 2006



RPA & TDDFT master equation


RPA & TDDFT master equation


A and B matrices


RPA & TDDFT master equation


How to solve?
CIS
Davidson diagonalization
Requires construction of A·X products for trial vectors X

RPA & TDDFT
Davidson-like diagonalization methods
Require construction of (A+B)·X, (A+B)·Y, (A-B)·X, (A-B)·Y products for trial vectors X and Y

For our purposes, it is sufficient to consider in details CIS only.


AO-driven CIS
Let us consider RHF-based CIS for simplicity Let us transform basis from determinants to SAPS (CSF) to solve separate problems for singlet states (and triplet states) and to avoid spin-orbitals. A·X = jbAia,jb·Xjb
Aia,jb=(a- i)ij ab + ...
The most efficient way to evaluate this part of the matrixvector product is to use MO basis.

Aia,jb=... + (ia|jb) + exchange-related terms
The most efficient way to calculate these parts of matrix-vector product (most time-consuming step) is to use AO basis: jb(ia|jb)·Xjb= jb pqrsCipCaqCjrCbs (pq|rs)·Xjb= pqCipCaqrs (pq|rs)·(jbCjrCbsXjb)


AO-driven CIS benefits
Does not require integral transformation
applicable simplifies to very large systems use of molecular symmetry

Is based on the construction of Fock-like matrices
All known approaches used to solve large-scale SCF problems can be also used in the case of CIS
Linear scaling
· QFMM · Linear exchange


Main CIS and SCF differences
CIS operates with non-symmetrical (square) density-like and Fock-like matrices CIS results in non totally-symmetric density and Fock-like matrices. The symmetry is defined by the symmetry of the target CIS state


Reminiscence. Use of molecular point group symmetry in SCF methods
At least two different ways:
Use of symmetrized AO basis functions and 2-e integrals Use of petite AO integral list:
Construction of the skeleton Fock matrix Projection/symmetrization to the totally symmetric irreducible representation at the end


Skeleton matrix symmetrization


Skeleton matrix symmetrization


Terminology


Gathering things together
Now we have all the tools to construct algorithm for (any symmetry) Fock-like matrix assembly using petite integral list Let us consider some 2-d irrep. E without the loss of generality. We are interested in getting CIS states transforming, say, as the first row of E only, as their counterparts can be constructed (if needed at all) after diagonalization of the CIS Hamiltonian using shift operators.


Strategy
PrsE,2
shift

Fock matrix construction, petite list

F'

shift

pq

F

'

pq

X

jb

E,1

PrsE,1

projection

Fpq

+ Fpq

Fpq

E,1

A·X

E,1


The only nontrivial (?) point
Shift and projection operators for densitylike matrices are not the same as for Focklike matrices!
Fock-like matrices:
R(g)

Density-like matrices:
R(g) should be replaced by R+(g-1)


Thank you for your attention!