Hybrid correlation models based on active-space partitioning: correcting second-order Moller-Plesset perturbation theory for bond-breaking reactions

Moller-Plesset second-order (MP2) perturbation theory breaks down at molecular geometries which are far away from equilibrium. We decompose the MP2 energy into contributions from different orbital subspaces and show that the divergent behavior of the MP2 energy comes from the excitations located within a small (or sometimes even the minimal) active space. The divergent behavior of the MP2 energy at large interfragment distances may be corrected by replacing a small number of terms by their more robust counterparts from coupled-cluster (CCSD) theory. We investigated several schemes of such a substitution, and we find that a coupling between the active-space CCSD and the remaining MP2 amplitudes is necessary to obtain the best results. This naturally leads us to an approach which has previously been examined in the context of cost-saving approximations to CCSD for equilibrium properties by Nooijen [J. Chem. Phys. 111, 10815 (1999)]. The hybrid MP2-CCSD approach, which has the same formal scaling as conventional MP2 theory, provides potential curves with a correct shape for bond-breaking reactions of BH, CH(4), and HF. The error of the MP2-CCSD method (measured against full configuration-interaction data) is smaller than that of MP2 at all interfragment separations and is qualitatively similar to that of full CCSD.     

Polymer density functional theory approach based on scaling second-order direct correlation function

A second-order direct correlation function (DCF) from solving the polymer-RISM integral equation is scaled up or down by an equation of state for bulk polymer, the resultant scaling second-order DCF is in better agreement with corresponding simulation results than the un-scaling second-order DCF. When the scaling second-order DCF is imported into a recently proposed LTDFA-based polymer DFT approach, an originally associated adjustable but mathematically meaningless parameter now becomes mathematically meaningful, i.e., the numerical value lies now between 0 and 1. When the adjustable parameter-free version of the LTDFA is used instead of the LTDFA, i.e., the adjustable parameter is fixed at 0.5, the resultant parameter-free version of the scaling LTDFA-based polymer DFT is also in good agreement with the corresponding simulation data for density profiles. The parameter-free version of the scaling LTDFA-based polymer DFT is employed to investigate the density profiles of a freely jointed tangent hard sphere chain near a variable sized central hard sphere, again the predictions reproduce accurately the simulational results. Importance of the present adjustable parameter-free version lies in its combination with a recently proposed universal theoretical way, in the resultant formalism, the contact theorem is still met by the adjustable parameter associated with the theoretical way.     

Explicitly correlated local second-order perturbation theory with a frozen geminal correlation factor

The recently introduced MP2-R122*A(loc) and LMP2-R122*A(loc) methods are modified to use a short-range correlation factor expanded as a fixed linear combination of Gaussian geminals. Density fitting is used to reduce the effort for integral evaluation, and local approximations are introduced to improve the scaling of the computational resources with molecular size. The MP2-F122*A(loc) correlation energies converge very rapidly with respect to the atomic orbital basis set size. Already with the aug-cc-pVTZ basis the correlation energies computed for a set of 21 small molecules are found to be within 0.5% of the MP2 basis set limit. Furthermore the short-range correlation factor leads to an improved convergence of the resolution of the identity, and eliminates problems with long-range errors in density fitting caused by the linear r12 factor. The DF-LMP2-F122*A(loc) method is applied to compute second-order correlation energies for molecules with up to 49 atoms and more than 1600 basis functions.     

Dual-basis second-order Moller-Plesset perturbation theory: A reduced-cost reference for correlation calculations

The resolution-of-the-identity (RI) approximation has placed the onus of the cost of a second-order Moller-Plesset (MP2) calculation on the underlying self-consistent field (SCF) calculation for many moderately sized molecules. A dual-basis approach to the SCF calculation, based on previous methods demonstrated for density functional theory, is combined with RI-MP2 calculations, and small basis subsets for cc-pVTZ, cc-pVQZ, and 6-311++G(3df,3pd) are presented. These subsets provide time savings of greater than 90%, with negligible errors in absolute and relative energies, compared to the associated full-basis counterpart. The method is tested with a series of rotational barriers, relative conformational energies of alanine tetrapeptides, as well as the full G3/99 molecular set. RI-MP2 calculations on alanine octapeptides (40 heavy atoms, 3460 basis functions), using cc-pVQZ, are presented. Results improve upon previous methods that diagonalize the virtual space separately.     

Communication: extended multi-state complete active space second-order perturbation theory: energy and nuclear gradients

The extended multireference quasi-degenerate perturbation theory, proposed by Granovsky [J. Chem. Phys. 134, 214113 (2011)], is combined with internally contracted multi-state complete active space second-order perturbation theory (XMS-CASPT2). The first-order wavefunction is expanded in terms of the union of internally contracted basis functions generated from all the reference functions, which guarantees invariance of the theory with respect to unitary rotations of the reference functions. The method yields improved potentials in the vicinity of avoided crossings and conical intersections. The theory for computing nuclear energy gradients for MS-CASPT2 and XMS-CASPT2 is also presented and the first implementation of these gradient methods is reported. A number of illustrative applications of the new methods are presented.     

Approximate fourth-order perturbation theory of the electron correlation energy

An approximate fourth-order expression for the electron correlation energy in the Møller–Plesset perturbation scheme is proposed. It takes into account all the contributions to the fourthorder energy neglecting only those of the triple-substituted determinants. It is size consistent and correct to fourth order for an assembly of isolated two-electron systems. Illustrative calculations are reported for a series of small molecules.     

Calculation of correlation energy by many-body diagrammatic perturbation theory

The many-body diagrammatic perturbation theory is used for calculation of the correlation energy of closed-shell molecular systems. We apply Brueckner's concept of the two-particle renormalized interaction defined by a non-linear diagrammatic expression containing all possible (diagonal and/or non-diagonal) particle-particle, hole-hole and particle-hole intermediate elementary processes. Then, a “second-order” simple diagrammatic expression for the correlation energy can be formed, where the correlation energy is approximated by all the diagrams with biexcited intermediate states. An illustrative numerical application for the LiH molecule is presented. This article is dedicated to the memory of our friends and colleagues Dr. Jarka Surá and Dr. Marta Černayová, who tragically died in July 1976.     

Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations

Recently, three of us have proposed a method [Phys. Rev. Lett. 91, 33201 (2003)] for an accurate calculation of the dispersion energy utilizing frequency-dependent density susceptibilities of monomers obtained from time-dependent density-functional theory (DFT). In the present paper, we report numerical calculations for the helium, neon, water, and carbon dioxide dimers and show that for a wide range of intermonomer separations, including the van der Waals and short-range repulsion regions, the method provides dispersion energies with accuracies comparable to those that can be achieved using the current most sophisticated wave-function methods. If the dispersion energy is combined with (i) the electrostatic and first-order exchange interaction energies as defined in symmetry-adapted perturbation theory (SAPT) but computed using monomer Kohn-Sham (KS) determinants, and (ii) the induction energy computed using the coupled KS static response theory, (iii) the exchange-induction and exchange-dispersion energies computed using KS orbitals and orbital energies, the resulting method, denoted by SAPT(DFT), produces very accurate total interaction potentials. For the helium dimer, the only system with nearly exact benchmark values, SAPT(DFT) reproduces the interaction energy to within about 2% at the minimum and to a similar accuracy for all other distances ranging from the strongly repulsive to the asymptotic region. For the remaining systems investigated by us, the quality of the SAPT(DFT) interaction energies is so high that these energies may actually be more accurate than the best available results obtained with wave-function techniques. At the same time, SAPT(DFT) is much more computationally efficient than any method previously used for calculating the dispersion and other interaction energy components at this level of accuracy.     

Undoing static correlation: long-range charge transfer in time-dependent density-functional theory

Long-range charge-transfer excited states are notoriously badly underestimated in time-dependent density-functional theory (TDDFT). We discuss how exact TDDFT captures charge transfer between open-shell species: in particular, the role of the step in the ground-state potential, and the severe frequency dependence of the exchange-correlation kernel. An expression for the latter is derived, that becomes exact in the limit that the charge-transfer excitations are well separated from other excitations. The exchange-correlation kernel has the task of undoing the static correlation in the ground state introduced by the step, in order to accurately recover the physical charge-transfer states.     

Different forms of perturbation theory for the calculation of the correlation energy

Different forms of perturbation theory for the calculation of correlation energy in both closed-and open-shell systems are discussed. For closed-shell systems, Epstein–Nesbet perturbation theory is compared with Møller–Plesset (MP ) perturbation theory based on canonical Hartree–Fock orbitals and with MP theory based on internally consistent SCF orbitals. The traditional MP theory gives superior results despite its use of an inferior zeroth-order Hamiltonian. This behavior is rationalized in terms of the larger denominators present in the traditional MP theory. These conclusions are used to support the restricted open-shell perturbation methods proposed recently by Murray and Davidson, and these new methods are compared with spin-restricted Epstein–Nesbet theory and the unrestricted MP (UMP ) approach. © 1992 John Wiley & Sons, Inc.     

Variational energy derivatives and perturbation theory

In this note we discuss the variational forms of the energy derivatives and the method of obtaining them. We show that perturbation theory can be formulated in terms of stationary energy derivatives and that this formulation extends the idea of the Hylleraas functional to infinite order and excited states.     

Second-order Kohn-Sham perturbation theory: correlation potential for atoms in a cavity

Second-order perturbation theory based on the Kohn-Sham Hamiltonian leads to an implicit density functional for the correlation energy E(c) (MP2), which is explicitly dependent on both occupied and unoccupied Kohn-Sham single-particle orbitals and energies. The corresponding correlation potential v(c) (MP2), which has to be evaluated by the optimized potential method, was found to be divergent in the asymptotic region of atoms, if positive-energy continuum states are included in the calculation [Facco Bonetti et al., Phys. Rev. Lett. 86, 2241 (2001)]. On the other hand, Niquet et al., [J. Chem. Phys. 118, 9504 (2003)] showed that v(c) (MP2) has the same asymptotic -alpha(2r(4)) behavior as the exact correlation potential, if the system under study has a discrete spectrum only. In this work we study v(c) (MP2) for atoms in a spherical cavity within a basis-set-free finite differences approach, ensuring a completely discrete spectrum by requiring hard-wall boundary conditions at the cavity radius. Choosing this radius sufficiently large, one can devise a numerical continuation procedure which allows to normalize v(c) (MP2) consistent with the standard choice v(c)(r-->infinity)=0 for free atoms, without modifying the potential in the chemically relevant region. An important prerequisite for the success of this scheme is the inclusion of very high-energy virtual states. Using this technique, we have calculated v(c) (MP2) for all closed-shell and spherical open-shell atoms up to argon. One finds that v(c) (MP2) reproduces the shell structure of the exact correlation potential very well but consistently overestimates the corresponding shell oscillations. In the case of spin-polarized atoms one observes a strong interrelation between the correlation potentials of the two spin channels, which is completely absent for standard density functionals. However, our results also demonstrate that E(c) (MP2) can only serve as a first step towards the construction of a suitable implicit correlation functional: The fundamental variational instability of this functional is recovered for beryllium, for which a breakdown of the self-consistent Kohn-Sham iteration is observed. Moreover, even for those atoms for which the self-consistent iteration is stable, the results indicate that the inclusion of v(c) (MP2) in the total Kohn-Sham potential does not lead to an improvement compared to the complete neglect of the correlation potential.     

A two‐scale approach to electron correlation in multiconfigurational perturbation theory

We present a new approach for the calculation of dynamic electron correlation effects in large molecular systems using multiconfigurational second‐order perturbation theory (CASPT2). The method is restricted to cases where partitioning of the molecular system into an active site and an environment is meaningful. Only dynamic correlation effects derived from orbitals extending over the active site are included at the CASPT2 level of theory, whereas the correlation effects of the environment are retrieved at lower computational costs. For sufficiently large systems, the small errors introduced by this approximation are contrasted by the substantial savings in both storage and computational demands compared to the full CASPT2 calculation. Provided that static correlation effects are correctly taken into account for the whole system, the proposed scheme represent a hierarchical approach to the electron correlation problem, where two molecular scales are treated each by means of the most suitable level of theory. © 2014 Wiley Periodicals, Inc.     

Excess energy and equation of state of fluids with hard-core potential models from a second-order Monte Carlo perturbation theory

Monte Carlo simulations in the NVT ensemble of the reference hard-sphere fluid have been performed to obtain the “exact” first- and second-order terms in the inverse temperature expansion of the free energy of fluids with hard-core potentials. The results have been used to obtain parametrizations of the free energy of fluids with Sutherland potentials with variable range as well as for a fluid with a hard-core Lennard–Jones potential. The results for the excess energy and the equation of state are compared with simulation data available in the literature for these fluids.     

Near-degeneracy corrections for second-order perturbation theory: comparison of two approaches

 We compare two approximate perturbation schemes which were developed recently to deal with the (quasi)degeneracy problem in many-body perturbation theory. We conclude that although the two methods were introduced on quite different theoretical grounds, their performances are quite similar, and present an improvement over traditional perturbation theory. Both methods are cheap in computation time, but cannot compete in accuracy with more sophisticated schemes such as complete-active-space perturbation theory or dressed particle theories. Received: 1 August 2000 / Accepted: 2 August 2000 / Published online: 19 January 2001     

Kohn-Sham perturbation theory: simple solution to variational instability of second order correlation energy functional

The orbital-dependent correlation energy functional resulting from second order Kohn-Sham perturbation theory leads to atomic correlation potentials with correct shell structure and asymptotic behavior. The absolute magnitude of the exact correlation potential, however, is greatly overestimated. In addition, this functional is variationally instable, which shows up for systems with nearly degenerate highest occupied and lowest unoccupied levels like Be. In this contribution we examine the simplest resummation of the Kohn-Sham perturbation series which has the potential to resolve both the inaccuracy and the instability problem of the second order expression. This resummation includes only the hole-hole terms of the Epstein-Nesbet series of diagrams, which has the advantage that the resulting functional is computationally as efficient as the pure second order expression. The hole-hole Epstein-Nesbet functional is tested for a number of atoms and ions. It is found to reproduce correlation and ground state energies with an accuracy comparable to that of state-of-the-art generalized gradient approximations. The correlation potential, on the other hand, is dramatically improved compared to that obtained from generalized gradient approximations. The same applies to all quantities directly related to the potential, as, for instance, Kohn-Sham eigenvalues and excitation energies. Most importantly, however, the hole-hole Epstein-Nesbet functional turned out to be variationally stable for all neutral as well as all singly and doubly ionized atoms considered so far, including the case of Be.     

Extended multi-configuration quasi-degenerate perturbation theory: the new approach to multi-state multi-reference perturbation theory

The distinctive desirable features, both mathematically and physically meaningful, for all partially contracted multi-state multi-reference perturbation theories (MS-MR-PT) are explicitly formulated. The original approach to MS-MR-PT theory, called extended multi-configuration quasi-degenerate perturbation theory (XMCQDPT), having most, if not all, of the desirable properties is introduced. The new method is applied at the second order of perturbation theory (XMCQDPT2) to the 1(1)A(')-2(1)A(') conical intersection in allene molecule, the avoided crossing in LiF molecule, and the 1(1)A(1) to 2(1)A(1) electronic transition in cis-1,3-butadiene. The new theory has several advantages compared to those of well-established approaches, such as second order multi-configuration quasi-degenerate perturbation theory and multi-state-second order complete active space perturbation theory. The analysis of the prevalent approaches to the MS-MR-PT theory performed within the framework of the XMCQDPT theory unveils the origin of their common inherent problems. We describe the efficient implementation strategy that makes XMCQDPT2 an especially useful general-purpose tool in the high-level modeling of small to large molecular systems.