Dynamical Correlation Energy Research Articles

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

Hartree−Fock−Heitler−London Method. 2. First and Second Row Diatomic Hydrides

The Hartree-Fock-Heitler-London, HF-HL, method is a new ab initio approach which variationally combines the Hartree-Fock, HF, and the Heitler-London, HL, approximations, yielding correct dissociation products. Furthermore, the new method accounts for nondynamical correlation and explicitly considers avoided crossing. With the HF-HL model we compute the ground-state potential energy curves for H2 [1Sigma+g], LiH [X 1Sigma+], BeH [2Sigma+], BH [1Sigma+], CH [2Pi], NH [3Sigma-], OH [2Pi], and FH [1Sigma+], obtaining in average 80% of the experimental binding energy with a correct representation of bond breaking. Inclusion of ionic configurations improves the computed binding energy. The computed dipole moment is in agreement with laboratory data. The dynamical and nondynamical correlation energies for atomic and molecular systems with 2-10 electrons are analyzed. For BeH the avoided crossing of the two lowest [2Sigma+] states is considered in detail. The HF-HL function is proposed as the zero-order reference wave function for molecular systems. To account for the dynamical correlation energy a post-HF-HL technique based on multiconfiguration expansions is presented. We have computed the potential energy curves for H2 [1Sigma+g], HeH [2Sigma+], LiH [X1Sigma+], LiH [A1Sigma+], and BeH [2Sigma+]. The corresponding computed binding energies are 109.26 (109.48), 0.01 (0.01), 57.68 (58.00), 24.19 (24.82), and 49.61 (49.83) kcal/mol, with the experimental values given in parentheses. The corresponding total energies are -1.1741, -3.4035, -8.0695, -7.9446, and -15.2452 hartrees, respectively, the best ab initio variational published calculations, H2 excluded.

Scaling Dynamical Correlation Energy from Density Functional Theory Correlation Functionals

The scaling of dynamical correlation energy in molecules obtained by the correlation functionals of density functional theory (DFT) is examined. The approach taken is very similar to the scaled external correlation method of Brown and Truhlar but is based on the observation that DFT correlation functionals, especially the LYP, appear to represent the dynamical portion of the correlation energy in molecules. We examine whether higher accuracy in atomization energies can be gained by scaling without significant deterioration of the structural and spectroscopic properties of the molecules using four DFT functionals (BLYP, OLYP, B3LYP, and O3LYP) on 19 molecules including the six molecule AE6 database, the latter being representative of a much larger, 109 molecule training set. We show that, with molecule specific scale factors, nearly perfect agreement with experiment can be achieved in atomization energies without increasing the average errors in other molecular properties relative to the DFT calculation. We further show that it is possible to find optimal scale factors which reduce the mean unsigned error per bond to levels comparable to those of some multilevel multicoefficient methods.

Density-functional theory and hybrid density-functional theory continuum solvation models for aqueous and organic solvents: universal SM5.43 and SM5.43R solvation models for any fraction of Hartree-Fock exchange

Hybrid density functional theory, which is a combined Hartree–Fock and density functional method, provides a simple but effective way to incorporate nonlocal exchange effects and static and dynamical correlation energy into an orbital-based theory with affordable computational cost for many important problems of gas-phase chemistry. The inclusion of a reaction field representing an implicit solvent in a self-consistent hybrid density functional calculation provides an effective and efficient way to extend this approach to problems of liquid-phase chemistry. In previous work, we have parameterized several models based on this approach, and in the present article, we present several new parameterizations based on implicit solvation models SM5.43 and SM5.43R. In particular, we extend the applicability of these solvation models to several combinations of the MPWX hybrid-density functional with various one-electron basis sets, where MPWX denotes a combination of Barone and Adamo’s modified version of Perdew and Wang’s exchange functional, Perdew and Wang’s correlation functional, and a percentage X of exact Hartree–Fock exchange. SM5.43R parameter optimizations are presented for the MPWX/MIDI!, MPWX/MIDI!6D, and MPWX/6-31+G(d,p) combinations with X=0 (i.e., pure density functional theory), 25, 42.8, and 60.6, and for MPWX/6-31G(d) and MPWX/6-31+G(d), with X=0, 42.8, and 60.6; this constitutes a total of 18 new parameter sets. [Note that parameter optimizations using MPW25/6-31G(d) and MPW25/6-31+G(d) were carried out in a previous SM5.43R parameterization.] For each of the five basis sets, we found no significant loss in the accuracy of the model when parameters averaged over the four values of X are used instead of the parameters optimized for a specific value of X. Therefore for each of the five basis sets used here, the SM5.43R and SM5.43 models are defined to have a single parameter set that can be used for any value of X between 0 and 60.6. The new models yield accurate free energies of solvation for a broad range of solutes in both water and organic solvents. On the average, the mean-unsigned errors, as compared with those from experiment, of the free energies of solvation of neutral solutes range from 0.50 to 0.55 kcal/mol and those for ions range from 4.5 to 4.9 kcal/mol. Since the SM5.43R model computes the free energy of solvation as a sum of bulk-electrostatic and non-bulk-electrostatic contributions, it may be used for detailed analysis of the physical effects underlying a calculation of the free energy of solvation. Several calculations illustrating the partitioning of these contributions for a variety of solutes in n-hexadecane, 1-octanol, and water are presented.

Experimental and theoretical investigation of the A3Π–X3Σ−transition of NH/D–Ne

A study of NH/D-Ne was undertaken to investigate the structure of this complex and examine the ability of high-level theoretical methods to predict its properties. The A 3pi-X 3sigma- transition was characterized using laser induced fluorescence measurements. Results from theoretical calculations were used to guide the interpretation of the spectra. Two-dimensional potential energy surfaces were calculated using second-order multireference perturbation theory with large correlation consistent basis sets. The potential energy surfaces were used to predict the ro-vibronic structure of the A-X system. Calculated ro-vibronic energy level patterns could be recognized in the spectra but quantitative discrepancies were found. These discrepancies are attributed to incomplete recovery of the dynamical correlation energy.

Repulsive double many-body expansion potential energy surface for the reactions N(4S)+ H2⇌ NH(X3Σ–)+ H from accurate ab initio calculations

A single-sheeted DMBE potential energy surface is reported for the reactions N(4S)+H2<-->NH(X3Sigma-)+H based on a fit to accurate multireference configuration interaction energies. These have been calculated using the aug-cc-pVQZ basis set of Dunning and the full valence complete active space wave function as reference, being semi-empirically corrected by scaling the two-body and three-body dynamical correlation energies. The topographical features of the novel global potential energy surface are examined in detail, including a conical intersection involving the two first 4A'' potential energy surfaces which has been transformed into an avoided crossing in the present single-sheeted representation.

Role of differential correlation energy in core ionization of pyrrole and pyridine

Δ Hartree–Fock ( Δ HF) and Δ Kohn–Sham ( Δ KS) orbital theories are used to calculate C(1s) and N(1s) core electron binding energies and chemical shifts of pyrrole and pyridine molecules. As a result, combination of Δ KS and B3LYP functional appears as a very promising approach for the determination of the core excitation/ionization energies. For the carbon series in pyridine, electronic correlation is essential in correctly predicting the relative shifts among carbon atoms. A strong dynamical differential correlation energy for carbon and nitrogen has been revealed and can be related to the large magnitude of electronic transfer from ligands to the core ionized site.

Isomerizations of Bicyclo[2.1.0]pent-2-ene and Tricyclo[2.1.0.02,5]pentane into Cyclopenta-1,3-diene: A Computational Study by DFT and High-Level ab Initio Methods

The thermal isomerizations of bicyclopentene (bcp) and tricyclopentane (tcp) into cyclopentadiene (cp) are investigated by a combination of DFT, CASSCF, CASSCF-MP2, and CR-CCSD(T) methods. Coupled-clusters and B3LYP methods predicted the reaction enthalpies excellently whereas the MCSCF method worked well only when dynamic correlation energy was taken into account. Both processes are concerted, and the reaction paths pass through transition states with high biradical character. Measures of biradical character in DFT and ab initio methods are discussed. The activation enthalpy in the rearrangement of bcp into cp was predicted by the CR-CCSD(T) method to be 25.5 kcal/mol, in good agreement with experiment. The UB3LYP functional also performed well in this case despite the high spin contamination that was present in the singlet biradicaloid transition state. The reaction enthalpy for the conversion of tcp into cp was predicted to be −63.7 kcal/mol. The transition state involved in the isomerization of tcp wa...

The well-tempered auxiliary-field Monte Carlo.

The auxiliary-field Monte Carlo (AFMC) is a method for computing ground-state and excited-state energies and other properties of electrons in molecules. For a given basis set, AFMC is an approximation to full-configuration interaction and the accuracy is determined predominantly by an inverse temperature "beta" parameter. A considerable amount of the dynamical correlation energy is recovered even at small values of beta. Yet, nondynamical correlation energy is inefficiently treated by AFMC. This is because the statistical error grows with beta, warranting increasing amount of Monte Carlo sampling. A recently introduced multi-determinant variant of AFMC is studied, and the method can be tuned by balancing the sizes of the determinantal space and the beta-parameter with respect to a predefined target accuracy. The well-tempered AFMC is considerably more efficient than a naive AFMC. As a welcome "byproduct" low lying excitation energies of the molecule are supplied as well. We demonstrate the principles on dissociating hydrogen molecule and torsion of ethylene where we calculate the (unoptimized) torsional barrier and the vertical singlet-triplet splitting.

Static and dynamic correlations in semiconductor heterostructures

Using the quantum Singwi–Tosi–Land–Sjolander (QSTLS) theory we have studied many-body effects in the quasi-two-dimensional electron liquid in GaAs/Al x Ga 1− x As heterojunctions. The leakage of the charge distribution to the barrier region is included through a variational approach. We have calculated the pair correlation function, structure factor, dynamic local-field correction, plasmon dispersion and correlation energy for a wide range of electron and ionized acceptor densities. In the limit of the 2D electron fluid our results match closely with Monte-Carlo data and our dynamic local field factor (LFF) is similar to those obtained in earlier papers. We have shown that the effect of the finite width of the electron layer on the dynamic LFF and pair distribution function is remarkable and there are significant differences in the results of the QSTLS and classical STLS theories for a wide range of parameters.

Combining two-body density functionals with multiconfigurational wavefunctions: diatomic molecules

The MCSCF method provides a correct zero-order wavefunction for all regions of molecular potential energy surfaces. To obtain quantitative accuracy a proper treatment of the dynamic correlation problem must be implemented. Traditionally this has been achieved through multireference variants of perturbation theory, configuration interaction and coupled cluster theory. The computational cost of such techniques makes them prohibitive for all but the smallest molecular problems. Reported here is an investigation into the efficacy of two-body density functionals in providing the dynamic correlation energy for MCSCF reference states. Tests were made on the two-body density functionals of Colle and Salvetti (CS), Moscardó and San-Fabán (MSF), and Moscardó and Pérez-Jiménez (MPJS) in predicting the equilibrium bond lengths, harmonic frequencies and dissociation eneries of fifteen diatomic molecules (3B2, 3BN, 2BS, 1C2, 2CN, 1CO, 1F2, 1FC1, 1N2, 3NCl, 3O2, 1PN, 3Si2, 1SiO, 3SO) using full valence-shell CASSCF reference wavefunctions. Also studied were modifications of these functionals recently suggested by Miehlich, Stoll and Savin (MSS) and Gráfenstein and Cremer (GC). The results obtained show accuracy comparable with and typically superior to the popular Kohn-Sham BLW and B3LW methods. However, the latter methods are not applicable in all regions of a potential energy surface, and even predict incorrect ground states for some systems. The use of two-body density functionals with MCSCF reference states does not share this shortcoming.

Direct diabatization of electronic states by the fourfold way. II. Dynamical correlation and rearrangement processes

Diabatic representation of coupled potential energy surfaces and their scalar couplings provides a compact and convenient starting point for dynamics calculations carried out in either the adiabatic or diabatic representation. In a previous paper we presented a general, path-independent scheme, called the fourfold way, for calculating diabatic surfaces and their scalar couplings from adiabatic surfaces and electronic density matrices such that the manifold of diabatic states spans the variationally optimized space of a finite number of adiabatic states. In the present paper we extend that scheme in these ways: (1) We show how to include dynamical electronic correlation energy by multireference perturbation theory or configuration interaction based on a complete active reference space. (2) We present a more general strategy for treating rearrangements. (3) We present consistency criteria for testing the validity of the assumptions for a particular choice of reference geometries, diabatic molecular orbital (DMO) ordering, dominant configuration-state-function lists, and choice(s) for reference DMO(s) for systems involving rearrangements. The first extension is illustrated by multiconfiguration quasidegenerate perturbation theory (MC-QDPT) calculations on LiF, and all three extensions are illustrated by MC-QDPT calculations on the reaction Li(2 2S,2 2P)+HF→LiF+H.

Direct generation of local orbitals for multireference treatment and subsequent uses for the calculation of the correlation energy

We present a method that uses the one-particle density matrix to generate directly localized orbitals dedicated to multireference wave functions. On one hand, it is shown that the definition of local orbitals making possible physically justified truncations of the CAS (complete active space) is particularly adequate for the treatment of multireference problems. On the other hand, as it will be shown in the case of bond breaking, the control of the spatial location of the active orbitals may permit description of the desired physics with a smaller number of active orbitals than when starting from canonical molecular orbitals. The subsequent calculation of the dynamical correlation energy can be achieved with a lower computational effort either due to this reduction of the active space, or by truncation of the CAS to a shorter set of references. The ground- and excited-state energies are very close to the current complete active space self-consistent field ones and several examples of multireference singles and doubles calculations illustrate the interest of the procedure.

A realistic double many-body expansion potential energy surface for [formula omitted] from a multiproperty fit to accurate ab initio energies and vibrational levels

A single-valued double many-body expansion potential energy surface (DMBE I) recently obtained for the ground electronic state of the sulfur dioxide molecule by fitting correlated ab initio energies suitably corrected by scaling the dynamical correlation energy is now refined by fitting simultaneously available spectroscopic levels up to 6886 cm −1 above the minimum. The topographical features of the novel potential energy surface (DMBE II) are examined in detail, and the method is emphasized as a robust route to fit together state-of-the-art theoretical calculations and spectroscopic measurements using a single fully dimensional potential form.

Correlated description of multiple bonds using localized active orbitals

It is shown that the use of atomic-like active orbitals instead of symmetry-adapted ones is computationally interesting for the treatment of strongly correlated problems, when the zero-order description involves a large complete active space (CAS). Using localized active orbitals makes it possible to reduce the number of references to the leading valence bond configurations, and hence to achieve the calculation of the dynamical correlation energy for a lower cost. This benefit would be especially dramatic for remote active centers but it remains significant in the most difficult problem of strong multiple bonds, as illustrated on N 2 and Mo 2Cl 8 4−. The valence bond analysis of the wavefunction makes clear the physical content of the non-dynamical correlation.

On the correlation energy features in planar heteroatomic molecular systems

The correlation energy in planar heteroatomic open chain polyene systems involving N, O, and F atoms is considered by the CASSCF and CASPT2 methods employing a number of the cc-pVmZ (VDZ, VTZ, etc.) correlation consistent basis sets. A thorough study of the smallest molecules shows that the nondynamical correlation energy is virtually independent of the quality of the basis set. In contrast, the dynamical correlation energy is very sensitive to the basis set and, in estimating reliable dynamical correlation effects for larger systems, one has to rely on adequate extrapolation formulas to obtain the infinite basis set limit. We find that a method recently proposed by Truhlar offers economical yet reasonable estimates of the complete basis set results. Investigation of the sensitivity of the results to the choice of active space and the comparison to single reference MP2 calculations indicate that such extrapolations offer a good general method for saturating the basis set in multireference calculations. Thus a simple refinement of the conventional multireference coupled cluster method is proposed. It is also shown that both nondynamical and dynamical correlation energies follow very simple additivity rules in linear and branched planar chain heteroatomic polyenes, making possible their prediction in very large systems without calculation.

Dynamic correlation

From a knowledge of the Hartree-Fock and exact non-relativistic energies of atoms, the correlation energy Ec, as defined by Lowdin, may be calculated. For atoms this correlation is defined as dynamic correlation. The separate like-spin and unlike-spin contributions, Ecσσ, Ecαβ, may be calculated as a sum of pair energies from quantum chemistry; we have used the unrestricted Møller-Plesset second-order algorithm, and then scaled them to give Ec. These three values may also be computed using dynamic correlation functionals, with the Stoll partitioning. The VWN, LYP and P91 functionals were studied for the atoms from H to Ar. Although the total correlation energies of LYP and P91 are similar, only P91 gives a semi-sensible breakdown into the Ecσσ, and Ecαβ components. It is immediately apparent that a new functional, OPTC, derived from the P91 components as 0.6625 x Ecσσ, + 1. 1015 x Ecαβ is an improvement (its mean absolute error is only 0.006 Eh). Using the recently introduced improved exchange functional OPTX (obtained through a fit to the HF energies of atoms), Kohn-Sham calculations were performed on the atoms using the OPT(=OPTX + OPTC) functional. The total energies have a mean absolute error of 0.006 Eh. The study then moves to molecules. First it is shown that the dynamic correlation energy contribution to the dissociation energies is very similar (within 2kcalmol−1 in most cases), whether it is calculated with LYP, P91 or OPTC. A calculation is then made of the HF contribution, the dynamic contribution through OPTC and the left-right contribution through OPTX, to molecular binding. In many cases the sum agrees with the observed value, but in some cases the prediction is significantly in error, e.g. O2 is overbound by 10 kcal mol−1. Thus either OPTX or OPTC or both are inadequate. An attempt was made to determine improved local exchange and correlation functionals by fitting to both atomic and molecular data, but this was unsuccessful. The conclusion is that the method is close to the limit of accuracy achievable from separately optimized local exchange and correlation functionals. Finally, a new hybrid functional O3LYP, which is a substantial improvement on B3LYP, is presented.

Left-right correlation energy

We first attempt to determine a local exchange functional Ex[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. Ex[p is determined from p and |δp|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ i |2 improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local Ex[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.

Overestimation of the stability of the π-delocalized versus the σ-localized configuration in radicals by current density functionals: the case of vinylacyl radicals

Calculations with the density functional theory (DFT) method using the most popular functional, Becke's three parameter hybrid with the Lee, Yang and Parr correlation functional, predict the π-delocalized configuration of the vinylacyl radical, CH2=CH—C(•)=O, to be more stable than the σ-localized configuration in contrast with ab initio unrestricted quadratic configuration interaction with single and double excitations calculations as previously found for the isoelectronic vinyl radical, Y-C(•)=CH2, bearing π-type α substituents. Experimental evidence on the electronic configuration adopted by vinyl radicals is contrasting. In the present case comparison with experiment indicates firmly that the currently available density functionals overestimate the stability of π-delocalized versus σ-localized configurations in radicals since they favor the π configuration for the γ-methylvinylacyl radical, CH3—CH=CH—C(•)=O, in contrast with unequivocal electron spin resonance data. This failure is mainly due to an incorrect estimate of dynamic correlation energy with DFT functionals.

Direct iterative solution of the generalized Bloch equation. III. Application to H2-cluster models

A state-selective multiconfigurational single-reference method that was outlined in the preceding paper of this series (H. Meißner and J. Paldus, J. Chem. Phys. 113, 2594 (2000); preceding paper), and is based on a quadratic iterative algorithm enabling the direct solution of the generalized Bloch equation, is applied to several model systems consisting of interacting hydrogen molecules, nowadays referred to as the H4, S4, and H8 models. These exactly solvable models are often used to test the efficacy of post-Hartree–Fock methods in their ability to recover both the dynamic and nondynamic correlation energies, since they enable a continuous variation of the degree of quasidegeneracy from the degenerate to nondegenerate limit by varying a single geometrical parameter, while simulating the dissociation of one or more single bonds. Various approximation schemes that were outlined in Part II, as well as their combinations, are tested and their performance evaluated. The size-extensivity deviations of those approximations that do not rely on the exponential cluster ansatz for the wave operator are also examined using larger hydrogen molecule clusters. It is shown that the so-called BQ4 approximation performs extremely well in all cases and even outperforms the externally corrected, reduced multireference (RMR) CCSD in the quasidegenerate region of geometries.