Valence Correlation Energy Research Articles

Evaluation of full valence correlation energies and gradients.

Complete-active-space self-consistent field (CASSCF) wave functions are central to understanding strongly correlated molecules as they capture the entirety of electronic interactions within a subset of the orbital space. The most interesting case for CASSCF is the full valence limit, where all bonding and an equal number of virtual orbitals are included in the active space, and no approximation is made in selecting the important valence orbitals or electrons. While conventional algorithms require exponential computational time to evaluate full valence CASSCF, this article shows that the method of increments can do the same with polynomial effort, in a new method denoted iCASSCF. The method of increments can also provide density matrices and other necessary ingredients for the construction of the nuclear gradient. These goals are met through a many-body expansion that breaks the problem into smaller pieces that are subsequently reassembled to form close approximations of conventional CAS results. Practical demonstrations on a number of medium-sized molecules, with up to 116 valence electrons correlated in 116 orbitals, show the power of this methodology.

Relaxing Constrained Amplitudes: Improved F12 Treatments of Orbital Optimization and Core–Valence Correlation Energies

We present a Lagrangian correction for the energy change upon releasing constraints imposed on coupled cluster amplitudes. We demonstrate that our correction (i) eliminates the systematic basis set error of the posthoc F12 treatment in orbital optimized methods and (ii) can be used to relax the F12 amplitudes and significantly reduce the sensitivity of the core-valence correlation energies to the exponent of the correlation factor, while retaining the low cost of the fixed amplitude approach.

Strongly localized approaches for delocalized systems. I. Ground state of linear polyenes

When approaching the theoretical description of the electronic structure of a molecule, instead of starting the construction of the electronic wave function from the self-consistent field single determinant, one may use a strongly localized determinant, corresponding to the Lewis picture of the molecule. Following this approach, the delocalization and the correlation effects are clearly identified and they can be treated simultaneously. This analysis, applied to the study of the ground state of π-conjugated systems, in particular to all-trans linear polyenes, shows the short-range character of the delocalization, essentially taking place between adjacent double bonds, and of the valence correlation energy, mainly coming from intra-double bond double excitations. This suggests that these systems can be described with a very limited model space, reduced to the corresponding excitations, the size of which scales linearly with the dimension of the system. Of course the effect of other excitations and of multiple excitations has to be taken into account at least in a perturbation scheme. The numerical studies on the first terms of the polyenes series show that the so-obtained energies are very close to those of the full π Complete Active Space Self-Consistent Field treatment. Moreover, one can correctly reproduce the dependence of the energy on the bond length alternation, thus showing the origin of the differences observed in these systems for the HF and CASSCF equilibrium geometries.

Statistical Electronic Structure Calibration Study of the CCSD(T*)-F12b Method for Atomization Energies

In the explicitly correlated CCSD(T)-F12b coupled cluster method only the singles and doubles component of the energy benefits from inclusion of terms involving the interelectronic distance. Consequently, only that component exhibits accelerated convergence with respect to the 1-particle basis set. The smaller perturbative triples component converges at the same rate as the corresponding piece in standard CCSD(T). With the alternative CCSD(T*)-F12b method the triples correlation energy is scaled up by the ratio of explicitly correlated to standard second-order perturbation theory correlation energies in an attempt to better approximate the basis set limit. An extensive and diverse 212 molecule collection of reference total atomization energies, developed with large basis sets (up to aug-cc-pV9Z in some cases) and standard CCSD(T), was used to calibrate the performance of CCSD(T*). Scaling of the (T) energy led to improved results relative to raw F12b values but only provided a statistical advantage over previously proposed complete basis set extrapolation techniques for the smallest basis sets. With larger sets, scaling (T) produced noticeably poorer results, sometimes by a factor of 2. In agreement with earlier studies, basis set extrapolated CCSD(T)-F12b was found to exhibit a systematic bias toward overestimating reference atomization energies with an error that increases with the magnitude of the valence correlation energy.

Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: The case of LiH

The atomic orbital basis set limit is approached in periodic correlated calculations for solid LiH. The valence correlation energy is evaluated at the level of the local periodic second order Møller-Plesset perturbation theory (MP2), using basis sets of progressively increasing size, and also employing "bond"-centered basis functions in addition to the standard atom-centered ones. Extended basis sets, which contain linear dependencies, are processed only at the MP2 stage via a dual basis set scheme. The local approximation (domain) error has been consistently eliminated by expanding the orbital excitation domains. As a final result, it is demonstrated that the complete basis set limit can be reached for both HF and local MP2 periodic calculations, and a general scheme is outlined for the definition of high-quality atomic-orbital basis sets for solids.

Fully automated incremental evaluation of MP2 and CCSD(T) core, core-valence and valence correlation energies

A recently proposed coupled cluster evaluation of core–core and core-valence correlation effects within the incremental scheme has been extended to perturbative treatments. The accuracy of the approach is demonstrated at the MP2 and CCSD(T) level of theory for various systems from different areas of chemistry, i.e. a binuclear titanium complex, a diallylmagnesium compound, a Hg 4 cluster and various hydration complexes of the sodium cation. Besides the convergence of individual correlation contributions arising from the core and/or valence electron systems the basis set dependence of the contributions was also monitored. Results within chemical accuracy of 1 kcal/mol in the total energies are typically obtained at third order of the incremental expansion. Furthermore a few reasonable simplifications of the incremental core-valence treatment are proposed, which allow a large number of individual calculations to be omitted a priori without a significant loss of accuracy.

The Ornstein–Zernike equation in molecular electronic structure theory

The Ornstein–Zernike equation expresses a simple relationship between direct and full correlation effects of a many-body system. Although it was introduced in the context of fluids, it can be derived through simple physical arguments that are equally applicable to the electrons in a molecule. Direct correlation effects account for the correlation between two particles at a time. If a simple model for direct correlation can be found, the Ornstein–Zernike equation can be used to convert it into a treatment of the full correlation effect. We show that the independent electron pair approximation (IEPA) is a reasonable model for the description of direct correlation effects. IEPA followed by our Ornstein–Zernike treatment is closer to CCSD(T) than is CCSD for valence correlation energies of small molecules.

Extrapolating MP2 and CCSD explicitly correlated correlation energies to the complete basis set limit with first and second row correlation consistent basis sets

Accurate extrapolation to the complete basis set (CBS) limit of valence correlation energies calculated with explicitly correlated MP2-F12 and CCSD(T)-F12b methods have been investigated using a Schwenke-style approach for molecules containing both first and second row atoms. Extrapolation coefficients that are optimal for molecular systems containing first row elements differ from those optimized for second row analogs, hence values optimized for a combined set of first and second row systems are also presented. The new coefficients are shown to produce excellent results in both Schwenke-style and equivalent power-law-based two-point CBS extrapolations, with the MP2-F12/cc-pV(D,T)Z-F12 extrapolations producing an average error of just 0.17 mE(h) with a maximum error of 0.49 for a collection of 23 small molecules. The use of larger basis sets, i.e., cc-pV(T,Q)Z-F12 and aug-cc-pV(Q,5)Z, in extrapolations of the MP2-F12 correlation energy leads to average errors that are smaller than the degree of confidence in the reference data (approximately 0.1 mE(h)). The latter were obtained through use of very large basis sets in MP2-F12 calculations on small molecules containing both first and second row elements. CBS limits obtained from optimized coefficients for conventional MP2 are only comparable to the accuracy of the MP2-F12/cc-pV(D,T)Z-F12 extrapolation when the aug-cc-pV(5+d)Z and aug-cc-pV(6+d)Z basis sets are used. The CCSD(T)-F12b correlation energy is extrapolated as two distinct parts: CCSD-F12b and (T). While the CCSD-F12b extrapolations with smaller basis sets are statistically less accurate than those of the MP2-F12 correlation energies, this is presumably due to the slower basis set convergence of the CCSD-F12b method compared to MP2-F12. The use of larger basis sets in the CCSD-F12b extrapolations produces correlation energies with accuracies exceeding the confidence in the reference data (also obtained in large basis set F12 calculations). It is demonstrated that the use of the 3C(D) Ansatz is preferred for MP2-F12 CBS extrapolations. Optimal values of the geminal Slater exponent are presented for the diagonal, fixed amplitude Ansatz in MP2-F12 calculations, and these are also recommended for CCSD-F12b calculations.

The ground-state potential curve of the beryllium dimer

The ground-state potential curve for the beryllium dimer is calculated using different methods. In the most accurate calculation a minimum of 0.9 kcal/mole is found at a distance of 4.9 a.u. Corrections for deficiencies in the basis set lead to a predicted well depth of 2.0 kcal/mole. Three classes of methods have been used—MCSCF, CI, and MBPT methods—and calculations were performed with two different basis sets. The largest basis set has 7s, 3p, and 2d contracted Gaussian functions where the 2d functions were optimized in the molecule. Nearly complete CI calculations with this basis set recovers 97% of the valence correlation energy. The main conclusion from the calculations with the different methods and basis sets is that it is necessary to compute practically all the valence shell correlation energy in order to obtain a reliable, qualitatively correct, potential curve for Be2. The best illustration of this point is that large MCSCF calculations with 1832 configurations, obtained from a complete distribution of 4 electrons in 20 orbitals, give qualitatively wrong potential curves. This is particularly remarkable since 98.8% of the valence correlation energy obtainable with the basis set is recovered.

Systematically convergent basis sets for explicitly correlated wavefunctions: The atoms H, He, B–Ne, and Al–Ar

Correlation consistent basis sets have been optimized for use with explicitly correlated F12 methods. The new sets, denoted cc-pVnZ-F12 (n=D,T,Q), are similar in size and construction to the standard aug-cc-pVnZ and aug-cc-pV(n+d)Z basis sets, but the new sets are shown in the present work to yield much improved convergence toward the complete basis set limit in MP2-F12/3C calculations on several small molecules involving elements of both the first and second row. For molecules containing only first row atoms, the smallest cc-pVDZ-F12 basis set consistently recovers nearly 99% of the MP2 valence correlation energy when combined with the MP2-F12/3C method. The convergence with basis set for molecules containing second row atoms is slower, but the new DZ basis set still recovers 97%-99% of the frozen core MP2 correlation energy. The accuracy of the new basis sets for relative energetics is demonstrated in benchmark calculations on a set of 15 chemical reactions.

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.

Nonlinear wave function expansions: A progress report

AbstractSome 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. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Combining explicitly correlated R12 and Gaussian geminal electronic structure theories

Explicitly correlated R12 methods using a single short-range correlation factor (also known as F12 methods) have dramatically smaller basis set errors compared to the standard wave function counterparts, even when used with small basis sets. Correlations on several length scales, however, may not be described efficiently with one correlation factor. Here the authors explore a more general MP2-R12 method in which each electron pair uses a set of (contracted) Gaussian-type geminals (GTGs) with fixed exponents, whose coefficients are optimized linearly. The following features distinguish the current method from related explicitly correlated approaches published in the literature: (1) only two-electron integrals are needed, (2) the only approximations are the resolution of the identity and the generalized Brillouin condition, (3) only linear parameters are optimized, and (4) an arbitrary number of (non-)contracted GTGs can appear. The present method using only three GTGs and a double-zeta quality basis computed valence correlation energies for a set of 20 small molecules only 2.2% removed from the basis set limit. The average basis set error reduces to 1.2% using a near-complete set of seven GTGs with the double-zeta basis set. The conventional MP2 energies computed with much larger quadruple, quintuple, and sextuple basis sets all had larger average errors: 4.6%, 2.4%, and 1.5%, respectively. The new method compares well to the published MP2-R12 method using a single Slater-type geminal (STG) correlation factor. For example, the average basis set error in the absolute MP2-R12 energy obtained with the exp(-r12) correlation factor is 1.7%. Correlation contribution to atomization energies evaluated with the present method and with the STG-based method only required a double-zeta basis set to exceed the precision of the conventional sextuple-zeta result. The new method is shown to always be numerically stable if linear dependencies are removed from the two-particle basis and the zeroth-order Hamiltonian matrix is made positive definite.

Accurate Thermochemical Properties for Energetic Materials Applications. I. Heats of Formation of Nitrogen-Containing Heterocycles and Energetic Precursor Molecules from Electronic Structure Theory

The heats of formation of 1H-imidazole, 1H-1,2,4-trizazole, 1H-tetrazole, CH3NO2, CH3N3, CH3NH2, CH2CHNO2, HClO4, and phenol, as well as cations and anions derived from some of the molecules have been calculated using ab initio molecular orbital theory. These molecules are important as models for compounds used for energetic materials synthesis. The predicted heats of formation of the heterocycle-based compounds are in excellent agreement with available experimental values and those derived from proton affinities and deprotonation enthalpies to <1 kcal/mol. The predicted value for the tetrazolium cation differs substantially from the experimental value, likely due to uncertainty in the measurement. The heats of formation of the nitro and amino molecules, as well as phenol/phenolate, also are in good agreement with the experimental values (<1.5 kcal/mol). The heat of formation of CH3N3 is predicted to be 72.8 kcal/mol at 298 K with an estimated error bar of +/-1 kcal/mol on the basis of the agreement between the calculated and experimental values for DeltaH(f)(HN3). The heat of formation at 298 K of HClO4 is -0.4 kcal/mol, in very good agreement with the experimental value, as well as a W2 literature study. An extrapolation of the CCSD(T)/aug-cc-pV(Q,5) energies was required to obtain this agreement. This result suggests that very large basis sets (> or =aug-cc-pV5Z) may be needed to fully recover the valence correlation energy contribution in compounds containing elements with high formal oxidation states at the central atom. In addition tight d functions are needed for the geometry predictions. Douglas-Kroll-Hess (DKH) scalar relativistic corrections for HClO4 and ClO4- at the MP2 level with correlation-consistent DKH basis sets were predicted to be large, likely due to the high formal oxidation state at the Cl.

The localizability of valence space electron–electron correlations in pair-based coupled cluster models

Inspired by the success of very restrictive local active space approaches like Generalized Valence Bond Perfect Pairing (GVB-PP) and Imperfect Pairing (IP), we investigate the localizability of valence correlation in the context of electron pair models. In particular, the role of two-center local excitations in local active space coupled cluster doubles models is examined in the context of a variety of chemical applications. The full two spatial center pairing model is found to recover the majority of the correlation effects missed by GVB-PP and IP, recovering up to 95% of the untruncated valence correlation energy, and could provide an inexpensive reference wave function for the treatment of additional electron–electron correlations. All of these models improve noticeably upon the Hartree–Fock wave function and can be computed for very large systems due to their only cubic computational cost. Their main weakness is a tendency to exhibit symmetry breaking in systems with multiple resonance structures.

Toward subchemical accuracy in computational thermochemistry: focal point analysis of the heat of formation of NCO and [H,N,C,O] isomers.

In continuing pursuit of thermochemical accuracy to the level of 0.1 kcal mol(-1), the heats of formation of NCO, HNCO, HOCN, HCNO, and HONC have been rigorously determined using state-of-the-art ab initio electronic structure theory, including conventional coupled cluster methods [coupled cluster singles and doubles (CCSD), CCSD with perturbative triples (CCSD(T)), and full coupled cluster through triple excitations (CCSDT)] with large basis sets, conjoined in cases with explicitly correlated MP2-R12/A computations. Limits of valence and all-electron correlation energies were extrapolated via focal point analysis using correlation consistent basis sets of the form cc-pVXZ (X=2-6) and cc-pCVXZ (X=2-5), respectively. In order to reach subchemical accuracy targets, core correlation, spin-orbit coupling, special relativity, the diagonal Born-Oppenheimer correction, and anharmonicity in zero-point vibrational energies were accounted for. Various coupled cluster schemes for partially including connected quadruple excitations were also explored, although none of these approaches gave reliable improvements over CCSDT theory. Based on numerous, independent thermochemical paths, each designed to balance residual ab initio errors, our final proposals are DeltaH(f,0) ( composite function )(NCO)=+30.5, DeltaH(f,0) ( composite function )(HNCO)=-27.6, DeltaH(f,0) ( composite function )(HOCN)=-3.1, DeltaH(f,0) ( composite function )(HCNO)=+40.9, and DeltaH(f,0) ( composite function )(HONC)=+56.3 kcal mol(-1). The internal consistency and convergence behavior of the data suggests accuracies of +/-0.2 kcal mol(-1) in these predictions, except perhaps in the HCNO case. However, the possibility of somewhat larger systematic errors cannot be excluded, and the need for CCSDTQ [full coupled cluster through quadruple excitations] computations to eliminate remaining uncertainties is apparent.

Accurate theoretical near-equilibrium potential energy and dipole moment surfaces of HgClO and HgBrO.

The complexes HgBrO and HgClO have been previously determined by ab initio methods to be strongly bound and were suggested to be important intermediates during mercury depletions events observed in the polar troposphere. In the present work accurate near-equilibrium potential energy surfaces (PESs) of these species are reported. The PESs are determined using accurate coupled cluster methods and a series of correlation consistent basis sets with subsequent extrapolation to the complete basis set limit. Additive corrections for both core-valence correlation energy and relativistic effects are also included. The anharmonic ro-vibrational spectra of HgBrO and HgClO have been calculated in variational calculations. Strong infrared band strengths are predicted for all fundamentals in these species. The spin-orbit splitting dominates over the vibronic coupling effect in both HgClO and HgBrO. The Renner-Teller vibronic energy levels corresponding to the bending mode of these molecules are calculated via perturbation theory.

Correlation energy estimates in periodic extended systems using the localized natural bond orbital coupled cluster approach

A new approach for the determination of correlation energies in periodic extended systems is proposed using the high transferability of amplitudes and integrals from natural bond orbital coupled cluster (NBO CC) calculations performed for small subunits. It is shown that the NBO CC calculations can in fact deliver detailed correlated wave function information for extended periodic systems. As an example we apply the ideas presented in this paper to determine an estimate for the valence correlation energy in diamond at the CCSD level.

Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited

Correlation consistent basis sets for accurately describing core–core and core–valence correlation effects in atoms and molecules have been developed for the second row atoms Al–Ar. Two different optimization strategies were investigated, which led to two families of core–valence basis sets when the optimized functions were added to the standard correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence–electron correlated calculations, i.e., for the core–core plus core–valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core–valence) correlation energy converges more slowly than the intrashell (core–core) correlation energy. Since the effect of including the core electrons on the calculation of molecular properties tends to be dominated by core–valence correlation effects, a second scheme for determining the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core–valence (intershell) correlation energy, except that a small amount of core–core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core–valence basis sets (cc-pwCVnZ), significantly improve the convergence of many molecular properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calculations on a series of second row homonuclear diatomic molecules (Al2, Si2, P2, S2, and Cl2), as well as on selected diatomic molecules involving first row atoms (CO, SiO, PN, and BCl). For the calculation of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.

A nonorthogonal approach to perfect pairing

We present an alternative formulation of perfect pairing (PP) aimed at giving a more faithful representation of the valence correlation energy of an arbitrary molecule. In the new theory, the occupied and virtual orbitals are nonorthogonal amongst themselves but orthogonal to each other. Whereas for the fully orthogonal version of PP one has the number of pairs equal to the number of occupied orbitals, the current formulation allows for an arbitrary number of pairs built from redundant orbitals. We propose setting the number of pairs equal to the number of valence orbitals in the molecule. Preliminary results indicate that the redundant formulation gives qualitatively improved results for delocalized systems such as benzene, while maintaining the attractive features of PP for localized systems.