Operator methods in full configuration interaction theory for molecular systems

We discuss modern trends in the theory and practice of full configuration interaction calculations. We pay the most attention to the wave operator method, in which the wave function is considered as the kernel of a many-particle operator. The corresponding operator equation, equivalent to the Schrödinger equation, automatically leads to a convenient matrix algorithm. We also discuss an alternative approach based on the pairing operator, generalizing the construction of the wave function in the method of one-particle spin-pairing amplitudes.Kharkov University. Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 27, No. 4, pp. 413–426, July–August, 1991. Original article submitted January 14, 1991.     

Cluster analysis of the full configuration interaction wave functions of cyclic polyene models

The first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI ) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.     

Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gas models

Using the homogeneous electron gas (HEG) as a model, we investigate the sources of error in the "initiator" adaptation to full configuration interaction quantum Monte Carlo (i-FCIQMC), with a view to accelerating convergence. In particular, we find that the fixed-shift phase, where the walker number is allowed to grow slowly, can be used to effectively assess stochastic and initiator error. Using this approach we provide simple explanations for the internal parameters of an i-FCIQMC simulation. We exploit the consistent basis sets and adjustable correlation strength of the HEG to analyze properties of the algorithm, and present finite basis benchmark energies for N = 14 over a range of densities 0.5 ≤ r(s) ≤ 5.0 a.u. A single-point extrapolation scheme is introduced to produce complete basis energies for 14, 38, and 54 electrons. It is empirically found that, in the weakly correlated regime, the computational cost scales linearly with the plane wave basis set size, which is justifiable on physical grounds. We expect the fixed-shift strategy to reduce the computational cost of many i-FCIQMC calculations of weakly correlated systems. In addition, we provide benchmarks for the electron gas, to be used by other quantum chemical methods in exploring periodic solid state systems.     

The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method

The recently proposed full configuration interaction quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancellation of the positive and negative particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics observed in simulations.     

Semiempirical andab initio calculations of the full configuration interaction using iterated Krylov’s spaces

The algorithm proposed previously for calculating the full configuration interaction using the variation matrix of the wave operator involves the numerical solution of the corresponding incomplete eigenvalue problem based on iterated Krylov’s subspaces. In practice, that means using the multistep gradient method as a special version of the Lanczos method. The high efficiency of this algorithm, which can readily be used in personal computer calculations, is proved by particular ab initio calculations of the full configuration interaction for the helium and beryllium atoms as well as by semiempirical calculations of π-shells for naphthalene and diphenylene. The algorithm is of particular assistance in obtaining numerous excited states, which are used for determining various spectral sums (polarizability, van der Waals interaction constants, and photoionization cross sections). Translated fromZhumal Struktumoi Khimii, Vol. 38, No. 1, pp. 14–22, January–February, 1997.     

Analytic expression of the second derivatives of electronic energy for full configuration interaction wave functions

Summary A new analytic second derivative expression of the electronic energy is derived for full configuration interaction (CI) wave functions. This formula is shown to be free from the derivative terms of both CI and MO coefficients. The second-order relationships between CI and MO coefficients for full CI wave functions are also presented.     

A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo

For the atoms with Z ≤ 11, energies obtained using the "initiator" extension to full configuration interaction quantum Monte Carlo (i-FCIQMC) come to within statistical errors of the FCIQMC results. As these FCIQMC values have been shown to converge onto FCI results, the i-FCIQMC method allows similar accuracy to be achieved while significantly reducing the scaling with the size of the Slater determinant space. The i-FCIQMC electron affinities of the Z ≤ 11 atoms in the aug-cc-pVXZ basis sets are presented here. In every case, values are obtained to well within chemical accuracy [the mean absolute deviation (MAD) from the relativistically corrected experimental values is 0.41 mE(h)], and significantly improve on coupled cluster with singles, doubles and perturbative triples [CCSD(T)] results. Since the only remaining source of error is basis set incompleteness, we have investigated using CCSD(T)-F12 contributions to correct the i-FCIQMC results. By doing so, much faster convergence with respect to basis set size may be achieved for both the electron affinities and the FCIQMC ionization potentials presented in a previous paper. With this F12 correction, the MAD can be further reduced to 0.13 mE(h) for the electron affinities and 0.31 mE(h) for the ionization potentials.     

Determination of spectroscopic terms in the LS-coupling scheme: a full configuration interaction space approach

This paper describes a procedure for determining the spectroscopic terms arising from an atomic or a linear molecular configuration. The method, based on a simple calculation of the dimensions of (L²,L_z)- and (Ŝ², Ŝ_z)-adapted full configuration interaction spaces, reduces the conventional situations of equivalent, non-equivalent (or both) electron configurations to one unique case. Some practical examples are considered.     

A graphical symmetric group approach for a spin adapted full configuration interaction: partitioning of a configuration graph into sets of closed-shell and open-shell graphs

We developed a spin adapted full configuration interaction (FCI) method which was expected to be effective for parallel processing. The graphical symmetric group approach (GSGA) was employed, where a configuration graph was partitioned into several sets of closed-shell and open-shell graphs. The configuration state functions (CSFs) bearing the same number of closed-shells and open-shells were assembled in a configuration group. The graphical approach provided a number to identify each CSF in a sequential order within the group. Combination of this partitioning and an intermediate configuration-driven algorithm in calculating the so-called σ vectors allowed us to use symbolic coupling constants. Furthermore, this combination made it easy to implement an efficient algorithm suitable to task-distributed parallel procedure for evaluating σ vectors. A program was written and some test calculations were carried out with high parallel efficiency. The largest size of FCI used 10 million CSFs (20 million determinants).     

Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods

The full configuration interaction quantum Monte Carlo (FCIQMC) method, as well as its "initiator" extension (i-FCIQMC), is used to tackle the complex electronic structure of the carbon dimer across the entire dissociation reaction coordinate, as a prototypical example of a strongly correlated molecular system. Various basis sets of increasing size up to the large cc-pVQZ are used, spanning a fully accessible N-electron basis of over 10(12) Slater determinants, and the accuracy of the method is demonstrated in each basis set. Convergence to the FCI limit is achieved in the largest basis with only O[10(7)] walkers within random errorbars of a few tenths of a millihartree across the binding curve, and extensive comparisons to FCI, CCSD(T), MRCI, and CEEIS results are made where possible. A detailed exposition of the convergence properties of the FCIQMC methods is provided, considering convergence with elapsed imaginary time, number of walkers and size of the basis. Various symmetries which can be incorporated into the stochastic dynamic, beyond the standard abelian point group symmetry and spin polarisation are also described. These can have significant benefit to the computational effort of the calculations, as well as the ability to converge to various excited states. The results presented demonstrate a new benchmark accuracy in basis-set energies for systems of this size, significantly improving on previous state of the art estimates.     

The direct configuration interaction method with a contracted configuration expansion

A variant of the configuration interaction method is outlined. There are four characteristic features of the method. First, the freezing, contraction of coefficients in the configuration expansion. In the present approach Rayleigh-Schrödinger perturbation theory has been used for this purpose. Second, the use of the direct CI-method to construct hamiltonian matrix elements between functions with contracted coefficients, in only one sequential read of the molecular integrals. Third, a diagonalization of the resulting small matrix. Fourth, an important computational advantage in that only one CI-vector is needed in core storage instead of two as in the usual direct CI-method. The method has been programmed for the case of all single and double replacements from a spin-unrestricted Hartree-Fock determinant. Test calculations indicate that usually less than 2% of the correlation energy is lost because of the contraction of the CI-expansion. A possibility to avoid the major part of the integral transformation and work directly with atomic basis functions is also discussed.     

Efficient methods for configuration interaction calculations

Advanced techniques are developed to provide efficient economic treatment of the large scale eigenvalue problem posed when configuration interaction is carried out on SCF basis sets of moderate size. When the characteristic properties of the hamiltonian matrix are examined in light of the type of solution required, partitioning of the configuration space is shown to result in an expansion of the problem about a limited core of states, where the small but cumulative interactions of vast regions of the remaining space are reduced to the form of an effective potential. With proper selection of the core, the evaluation of this potential can be readily and accurately truncated to a level involving minimum expenditure in time and effort. In particular only diagonal elements and a strip of the full CI matrix are required to achieve an accuracy of 1 – 5 kcal/mole with complete treatment for configuration spaces of order tens of thousands. In addition, a close look at current theory on the generation of matrix elements between spin symmetry adapted configurations leads to simplified expressions where the matrix elements are derived in the form of a weighted sum of molecular integrals in which the weighting coefficients represent the integrated value of the wavefunctions over spin coordinates. For typical cases of low multiplicity and limited numbers of open shells the list of unique parameters needed to generate all weights are shown to be readily stored as a program library. Actual times for matrix element generation are believed to be an order of magnitude faster than current techniques. Practical demonstration of the accuracy and efficiency of the method is provided by calculations on formaldehyde, water, and ethylene.     

Matrix elements in configuration interaction calculations

A method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.     

Nuclear-electronic orbital nonorthogonal configuration interaction approach

The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) approach is presented. In this framework, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear-electronic time-independent Schrodinger equation is solved with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wave function. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between the two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEO-full CI and Fourier grid calculations. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wave functions and the corresponding hydrogen tunneling splittings.     

Coupled-cluster method tailored by configuration interaction

A method is presented which combines coupled cluster (CC) and configuration interaction (CI) to describe accurately potential-energy surfaces (PESs). We use the cluster amplitudes extracted from the complete active space CI calculation to manipulate nondynamic correlation to tailor a single reference CC theory (TCC). The dynamic correlation is then incorporated through the framework of the CC method. We illustrate the method by describing the PESs for HF, H2O, and N2 molecules which involve single, double, and triple bond-breaking processes. To the dissociation limit, this approach yields far more accurate PESs than those obtained from the conventional CC method and the additional computational cost is negligible compared with the CC calculation steps. We anticipate that TCC offers an effective and generally applicable approach for many problems.