Perspective on "Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems"

An energy-based optimization method is presented for our recently developed nonlinear wave function expansion form for electronic wave functions. This expansion form is based on spin eigenfunctions, using the graphical unitary group approach (GUGA). The wave function is expanded in a basis of product functions, allowing application to closed-shell and open-shell systems and to ground and excited electronic states. Each product basis function is itself a multiconfigurational function that depends on a relatively small number of nonlinear parameters called arc factors. The energy-based optimization is formulated in terms of analytic arc factor gradients and orbital-level Hamiltonian matrices that correspond to a specific kind of uncontraction of each of the product basis functions. These orbital-level Hamiltonian matrices give an intuitive representation of the energy in terms of disjoint subsets of the arc factors, they provide for an efficient computation of gradients of the energy with respect to the arc factors, and they allow optimal arc factors to be determined in closed form for subspaces of the full variation problem. Timings for energy and arc factor gradient computations involving expansion spaces of Ͼ10 24 configuration state functions are reported. Preliminary convergence studies and molecular dissociation curves are presented for some small molecules.

Optimization of nonlinear wave function parameters

Shepard

Minkoff

2006

Nonlinear wave function expansions: A progress report

Shepard

Minkoff

Brozell

2007

Some recent progress is reported for a novel nonlinear expansion form for electronic wave functions. This expansion form is based on spin eigenfunctions using the Graphical Unitary Group Approach and the wave function is expanded in a basis of product functions, allowing application to closed and open shell systems and to ground and excited electronic states. Each product basis function is itself a multiconfigurational expansion that depends on a relatively small number of nonlinear parameters called arc factors. Efficient recursive procedures for the computation of reduced one-and two-particle density matrices, overlap matrix elements, and Hamiltonian matrix elements result in a very efficient computational procedure that is applicable to very large configuration state function (CSF) expansions. A new energy-based optimization approach is presented based on product function splitting and variational recombination. Convergence of both valence correlation energy and dynamical correlation energy with respect to the product function basis dimension is examined. A wave function analysis approach suitable for very large CSF expansions is presented based on Shavitt graph node density and arc density. Some new closed-form expressions for various Shavitt Graph and Auxiliary Pair Graph statistics are presented.

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S² eigenfunctions

Polyak

Bearpark

Robb

2017

Self Cite

We present an implementation of the spin-dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry-adapted functions.Using S 2 eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin-independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one-and two-electron charge densities. K E Y W O R D S configuration interaction, configuration state function, quantum chemistry, spin density, unitary group approach 1 | I N TR ODU C TI ONThe unitary group approach (UGA) in quantum chemistry was pioneered by Paldus in 1974, [1] with practical algorithms including the most famous graphical approach by Shavitt, [2,3] having been developed in the following years. It offered a simple and straightforward way of evaluating the spinless generator matrix elements (ME) using implicit Gelfand-Tsetlin basis [4,5] and was suitable for calculating many-electron wavefunction expectation values for spin-independent Hamiltonians. [6] The growing demand for relativisitc corrections and other spin dependent properties in wavefunction methods inspired the subsequent development of spin-dependent generator ME in 80-90s by the several groups of authors [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the latest work in this field is actually quite recent. [24] In particular, a theory for spin dependent operators formulated entirely within the unitary group formalism has been developed by Gould, Chandler, Paldus, and Battle. [7][8][9][10][11][12][13][14][15] As we shall discuss, the final formulae are easy to implement and are based on spin-independent U(n) and well-known U(2) generator ME in terms of the Gelfand-Tsetlin implicit basis. As a special, zero-order case, this theory provides a rather simple way for calculating spin density. [13,15,25] An obvious approach to spin density calculation with configuration interaction (CI) wavefunction would be to use the precomputed spindependent generator ME. The general single-particle reduced density operator (in a 2 3 2 matrix form) is given by [13] : qðr; r 0 Þ lm 5 X n i;j51