Use of perturbation methods for the study of configuration interaction effects

The second order correction to the energy of the ground state involves a quadruple summation over molecular orbitals. We show here that the effect of the triexcited configurations on a monoexcited state is cancelled by the effect of most of the diexcited states on the ground state. Thus the expression for the 2nd order correlated transition energies implies only triple summations over Molecular Orbitals. The singlet-triplet splitting is given by double summations. Some very simple rules are given for the choice of the finally useful configurations.

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Zusammenfassung Die BeitrÄge zweiter Ordnung des Grundzustandes ziehen eine vierfache Summation über MO's nach sich. Es wird nachgewiesen, da\ der Effekt von dreifach angeregten Konfigurationen auf einen einfach angeregten Zustand durch gegenseitige Eliminierung aufgehoben wird Ähnlich wie die Wirkung der meisten zweifach angeregten Konfigurationen auf den Grundzustand, so da\ sich nur eine dreifache Summation ergibt. Die Singulett-Triplett-Aufspaltung ist durch eine Doppelsumme gegeben. Es werden einige sehr einfache Regeln für die Wahl der schlie\lich benötigten Konfigurationen angegeben.

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Résumé Le calcul de l'énergie de corrélation au 2è ordre pour l'état fondamental implique une sommation quadruple portant sur les orbitales moléculaires. Nous démontrons dans cet article que les perturbations énergétiques d'un état monoexcité par les configurations «triexcitées» correspondent exactement à celles apportées au fondamental par la plupart des configurations «diexcitées». Ceci se traduit par un grand nombre de suppressions de termes dans l'expression des énergies de transition modifiées par la corrélation au 2è ordre, de sorte qu'il ne reste plus dans une telle expression que des sommations triples sur les orbitales moléculaires. De mÊme, la différence d'énergie singulet-triplet est donnée par des sommations doubles. Nous donnons quelques règles trés simples concernant le choix des configurations qui sont en fin de compte nécessaires.

     

The use of perturbation methods for the study of the effects of configuration interaction

The second order contribution to correlation energy of linear polyenes and polyacenes is studied in two partitions of the C.I. matrix, starting from delocalized Molecular Orbitals. When one uses the classical partition H=H _SCF+V, the correlation energy increases proportionnai to the number of electrons, quite independently of the shape of the molecule. Another partition, which insures the perturbation matrix to be zero-diagonal, gives a larger 2 ^nd order correlation energy; the difference between the 2 ^nd order contributions of these two expressions tends to a constant and is larger for a compact system than for a linear one. The dependence of the correlation energy to the values of bielectronic integrals used at short distances shows that it arises mainly from short range interactions.Some statistical models of C.I. matrices are proposed on the basis of the results obtained here. They give some interesting results for second order energies but do not seem to be satisfactory for higher orders.

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Zusammenfassung Der Beitrag zweiter Ordnung zur -Korrelationsenergie linearer Polyene und Polyacene wird mittels zweier Aufteilungen der CI-Matrix studiert, ausgehend von delokalisierten MO's. Wenn man die klassische Aufteilung H = H _SCF + V benutzt, nimmt die Korrelationsenergie proportional der Zahl der Elektronen zu, unabhÄngig von der Gestalt des Moleküls. Eine andere Aufteilung, bei der die Diagonalelemente der Störungsmatrix verschwinden, gibt einen grö\eren Beitrag zweiter Ordnung zur Korrelationsenergie. Die Differenz dieser beiden BeitrÄge geht gegen eine Konstante und ist für kompakte Systeme grö\er als für lineare Moleküle. Es wird gezeigt, da\ die Korrelationsenergie im wesentlichen auf Wechselwirkungen kurzer Reichweite zurückgeführt werden kann. Einige statistische Modelle für CI-Matrizen werden auf der Grundlage der hier erhaltenen Resultate vorgeschlagen. Sie scheinen für Energien zweiter Ordnung interessant zu sein, jedoch nicht für höhere Ordnungen.

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Résumé On étudie la contribution du 2è ordre à l'énergie de corrélation de polyènes et polyacènes linéaires pour deux partitions différentes de la matrice d'Interaction de Configuration (IC); on utilise des orbitales moléculaires délocalisées. Quand on emploie la partition classique H = H _SCF+V, l'énergie de corrélation au 2è ordre croÎt proportionnellement au nombre d'électrons, indépendamment de la forme de la molécule. Une autre partition, pour laquelle la matrice de l'opérateur de perturbation a ses éléments diagonaux nuls, donne une énergie de corrélation au 2è ordre plus grande; la différence entre ces deux contributions du 2è ordre tend vers une constante et est plus grande pour une molécule compacte que pour une molécule linéaire. La variation de cette énergie de corrélation (du 2è ordre) en fonction des paramètres utilisés pour les courtes distances montre que cette énergie provient surtout d'interactions à courte distance. On considère divers modèles statistiques de matrices d'IC fondés sur les résultats précédemment décrits. Ces modèles fournissent des résultats intéressants en ce qui concerne l'énergie de perturbation du 2è ordre, mais ils ne donnent pas satisfaction aux ordres supérieurs.

     

Iterative natural orbitals for configuration interaction using perturbation theory

Approximate natural orbitals are determined iteratively from CI expansions constructed using first-order perturbation theory in order to investigate the possibility of eliminating the complete transformation of MO integrals on each iteration. Results on LiH and H₂O are compared with fully variationally determined NO's to assess questions of convergence.     

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.     

Scaled second-order perturbation corrections to configuration interaction singles: efficient and reliable excitation energy methods

Two modifications of the perturbative doubles correction to configuration interaction with single substitutions (CIS(D)) are suggested, which are excited state analogues of ground state scaled second-order M?ller-Plesset (MP2) methods. The first approach employs two parameters to scale the two spin components of the direct term of CIS(D), starting from the two-parameter spin-component scaled (SCS) MP2 ground state, and is termed SCS-CIS(D). An efficient resolution-of-the-identity (RI) implementation of this approach is described. The second approach employs a single parameter to scale only the opposite-spin direct term of CIS(D), starting from the one-parameter scaled opposite-spin (SOS) MP2 ground state, and is called SOS-CIS(D). By utilizing auxiliary basis expansions and a Laplace transform, a fourth-order algorithm for SOS-CIS(D) is described and implemented. The parameters that describe SCS-CIS(D) and SOS-CIS(D) are optimized based on a training set that includes valence excitations of various organic molecules and Rydberg transitions of water and ammonia, and they significantly improve upon CIS(D) itself. The accuracy of the two methods is found to be comparable. This arises from a strong correlation between the same-spin and the opposite-spin portions of the excitation energy terms. The methods are successfully applied to the zincbacteriochlorin-bacteriochlorin charge-transfer transition, for which time-dependent density functional theory, with presently available exchange-correlation functionals, is known to fail. The methods are also successfully applied to describe various electronic transitions outside of the training set. The efficiency of the SOS-CIS(D) and the auxiliary basis implementation of CIS(D) and SCS-CIS(D) are confirmed with a series of timing tests.     

Application of perturbation theory in large configuration interaction calculations

Rayleigh-Schrödinger perturbation theory has been applied through fifth order in the energy, to the problem of estimating the roots of the secular equation in large configuration interaction calculations. The NO₂⁺, O₃ and H₂O molecules are used as test cases, with accuracy as good as 0.01 eV, with appropriate choice of zero order problem.     

Dissociation energies within selected configuration interaction and perturbation theory

Selected configuration interaction (CI) calculations and second-order perturbational theory are used to truncate systematically multireference single and double excitation CI (MRCI) expansions in the calculation of the bond dissociation energies of several systems like the single-bonded LiF molecule or the multiple-bonded N₂, NO and O₂ diatomic systems. The method is extended to compute the CH bond dissociation energy ofethene C₂H₄. It is shown how the proposed scheme (perturbation-selected MRCI (MRCI-PS)) is able to reproduce the accuracy of complete MRCI expansions with only a small number of configurations variationally evaluated.     

An application of perturbation theory ideas in configuration interaction calculations

Configuration interaction calculations of electronic wave functions for atoms and molecules have generally been limited to relatively small basis sets because of the exponential increase in the number of configurations as basis functions are added. While higher than quadruply excited configurations are of negligible importance in CI wave functions, it is shown that the effect of triple and quadruple excitation configurations can be substantially included even when the matrix elements between such configurations are neglected, leaving only their diagonal elements and the elements connecting them with the single and double excitations. This approximation is seen to be formally practically equivalent to a first-order perturbation expression for the wave function (second-order for the energy) based on an optimum linear combination of the zero, single, and double excitation configurations as the zero-order function. If suitable procedures are used, the amount of computational effort involved in such a calculation is roughly proportional to the fourth power of the number of basis functions employed, thus preventing the CI stage of the calculation from increasing in magnitude much faster than the stages involving the calculation and manipulation of the elementary integrals.     

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.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

Iterative calculation by perturbation of the configuration interaction for fundamental state and first excited states of MgO

For the lowest energy states of the MgO molecule, zero-order multiconfigurational wavefunctions are constructed by an iterative process, the remaining configuration interaction being treated by a Rayleigh—Schrödinger second-order perturbation. The calculated energies compare well with the experimental spectrum. The ground state appears to be a¹Σ⁺ state built essentially from the closed-shell SCF determinant and a di-excited (6σ)² → (7σ)² determinant.     

Multireference configuration interaction study of bromocarbenes

Multireference configuration interaction (MRCI) calculations of the lowest singlet X(1A') and triplet ?((3)A') states as well as the first excited singlet ?((1)A') state have been performed for a series of bromocarbenes: CHBr, CFBr, CClBr, CBr(2), and CIBr. The MRCI calculations were performed with correlation consistent basis sets of valence triple-ζ plus polarization quality, employing a full-valence active space of 18 electrons in 12 orbitals (12 and 9, respectively, for CHBr). Results obtained include equilibrium geometries and harmonic vibrational frequencies for each of the electronic states, along with ?((3)A') ← X((1)A') singlet-triplet gaps and ?((1)A') ← X((1)A') transition energies. Comparisons have been made with previous computational and experimental results where available. The MRCI calculations presented in this work provide a comprehensive series of results at a consistent high level of theory for all of the bromocarbenes.     

Third- and fourth-order perturbation corrections to excitation energies from configuration interaction singles

Complete third-order and partial fourth-order Rayleigh-Schrodinger perturbation corrections to excitation energies from configuration interaction singles (CIS) have been derived and termed CIS(3) and CIS(4)(P). They have been implemented by the automated system TENSOR CONTRACTION ENGINE into parallel-execution programs taking advantage of spin, spatial, and index permutation symmetries and applicable to closed- and open-shell molecules. The consistent use of factorization, first introduced by Head-Gordon et al. in the second-order correction to CIS denoted CIS(D), has reduced the computational cost of CIS(3) and CIS(4)(P) from O(n(8)) and O(n(6)) to O(n(6)) and O(n(5)), respectively, with n being the number of orbitals. It has also guaranteed the size extensivity of excited-state energies of these methods, which are in turn the sum of size-intensive excitation energies and the ground-state energies from the standard M?ller-Plesset perturbation theory at the respective orders. The series CIS(D), CIS(3), and CIS(4)(P) are usually monotonically convergent at values close to the accurate results predicted by coupled-cluster singles and doubles (CCSD) with a small fraction of computational costs of CCSD for predominantly singly excited states characterized by a 90%-100% overlap between the CIS and CCSD wave functions. When the overlap is smaller, the perturbation theory is incapable of adequately accounting for the mixing of the CIS states through higher-than-singles sectors of the Hamiltonian matrix, resulting in wildly oscillating series with often very large errors in CIS(4)(P). Hence, CIS(3) and CIS(4)(P) have a rather small radius of convergence and a limited range of applicability, but within that range they can be an inexpensive alternative to CCSD.     

Two-center orbital basis for configuration interaction study of diatomics

Two-center orbitals expressed in elliptic coordinates are used in a configuration interaction study of the hydrogen molecule and of the system He₂. A minimization of the energy is performed with respect to the nuclear charges defining each two-center orbital, either occupied or unoccupied, in the dominant configuration.     

Multiple substitution effects in configuration interaction calculations

The common way of approximating multiple substitution effects by the unlinked cluster contribution based on the linked cluster theorem is investigated. Formally the analysis is done in a configuration interaction language without reference to diagrams. Using the linked cluster theorem a series expansion is then developed for the correction of a CI calculation, including double replacements only, for the erratic treatment of unlinked clusters. The first term in this series is closely related to what is generally referred to as Davidson's correction. The next term is proportional to the square of the correlation energy and is also very easy to calculate. Finally numerical examples are given illustrating the accuracy of the different approximations.     

Comparison of correlation effects on the inversion barriers in CF3 and NF3+ from perturbation theory and configuration interaction

The barriers to inversion in NF₃⁺ and CF₃ have been studied using fourth-order unrestricted Møller-Plesset perturbation theory and configuration interaction with all single and double excitations relative to UHF wavefunctions. The barriers calculated by the two methods are in agreement to within 1 kcal/mol. Correlation effects are found to increase the barrier in NF₃⁺ while in the isoelectronic CF₃ species correlation decreases the barrier.     

Quasidegenerate scaled opposite spin second order perturbation corrections to single excitation configuration interaction

Scaled opposite spin (SOS) second order perturbative corrections to single excitation configuration interaction (CIS) are extended to correctly treat quasidegeneracies between excited states. Two viable methods, termed as SOS-CIS(D(0)) and SOS-CIS(D(1)), are defined, implemented, and tested. Each involves one empirical parameter (plus a second for the SOS-MP2 ground state), has computational cost that scales with the fourth power of molecule size, and has storage requirements that are cubic, with only quantities of the rank of single excitations produced and stored during iterations. Tests on a set of low-lying adiabatic valence excitation energies and vertical Rydberg excitations of organic and inorganic molecules show that the empirical parameter can be acceptably transferred from the corresponding nondegenerate perturbation theories without any further fitting. Further tests on higher excited states show that the new methods correctly perform for surface crossings for which nondegenerate approaches fail. Numerical results show that SOS-CIS(D(0)) appears to treat Rydberg excitations in a more balanced way than SOS-CIS(D(1)) and is, therefore, likely to be the preferred approach. It should be useful for exploring excited state geometries, transition structures, and conical intersections for states of medium to large organic molecules that are dominated by single excitations.     

Molecular response properties from explicitly time-dependent configuration interaction methods

In this paper we report the calculation of molecular electric response properties with the help of explicitly time-dependent configuration interaction (TD-CI) methods. These methods have the advantage of being applicable (within the limitations of the time-dependent Schrodinger equation) to time-dependent perturbations of arbitrary shape and strength. Three variants are used to solve the time-dependent electronic Schrodinger equation, namely, the TD-CIS (inclusion of single excitations only), TD-CISD (inclusion of single and double excitations), and TD-CIS(D) (single excitations and perturbative treatment of double excitations) methods and applied for illustration to small molecules, H(2) and H(2)O. In the calculation, slowly varying off-resonant electric fields are applied to the molecules and linear (polarizabilities) and nonlinear (hyperpolarizabilities, harmonic generation) response properties are determined from the time-dependent dipole moments.     

Multi-component molecular orbital study of isotope effects on lithium hydride molecules with the configuration interaction scheme

The multi-component molecular orbital method, which can take account of the quantum effect of the electrons and nuclei, is applied to the calculation of lithium hydride isotope species with the configuration interaction (CI) scheme. The optimum basis set functions for quantum nuclei are proposed by the fully variational procedure under single electronic–single nuclear excitation CI level. The average internuclear distances and dipole moments for isotopic lithium hydride molecules calculated with small basis functions are reasonable agreement with the corresponding experimental values.