Many-body perturbation theory applied to hydrogen fluoride

Pair contributions to the electronic correlation energy of the hydrogen fluoride molecule are computed using many-body perturbation theory (MBPT) with a basis set of Slater orbitals on both atomic centers. Summation of all two-body diagrams through third order gives ≈ 93% of the total correlation energy at the equilibrium internuclear separation. However, including certain higher-order effects, by means of denominator shifts, gives a correlation energy of ?0.3691 hartree, or ≈ 97% of the correlation energy.     

Molecular calculations for HeH<Superscript>+</Superscript> with two-center correlated orbitals

A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH⁺ molecular ion. The wavefunctions were constructed from the exact solution of the Schrödinger equation for the HeH⁺⁺ problem in prolate-spheroidal coordinates taken together with a Hylleraas type correlation factor. With a simple single term wavefunction, we obtained ground state energy of ?2.95308691 hartree without any variational parameters in the calculation. When a two-configuration-state wavefunction was used and effective charges were allowed to be adjusted, we found an energy of ?2.97384868 hartree, which is to be compared with ?2.97869074 hartree obtained by an 83 term configuration interaction wavefunction or ?2.97364338 hartree by an ab initio calculation (at the MP4(SDQ)/6-311++G(3df, 3dp) level) using the well-known “canned” code.     

Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Comparison between the symmetry-correlation examination and the perturbation method

A comparison between the construction of symmetry-correlation diagrams and the perturbation method for studying chemical reactions is carried out. The perturbation method consists of decomposing the system Hamiltonian H into a sum, H = H₀ + H′. Various symmetry correlation schemes appearing in the literature may be explained by the nonuniqueness of the decomposition scheme. All symmetry selection rules may be viewed as the varieties. By examining the symmetry-correlation diagrams, processes under investigation may be called “forbidden” or “allowed,” depending on the topological feature. Of particular importance is the topology associated with the “avoided crossing.” By making the comparison, we can establish the correspondence of the two methods and conclude that the perturbation order furnishes the origin of the “forbiddenness” of a process.     

Bond types of molecular orbitals and the photoelectron spectrum

The vibrational structures of the photoelectron spectra for diatomic molecules can be accounted for in terms of the slope of the orbital energy curve in the conventional correlation diagram with respect to internuclear distance. The vibrational structures of the photoelectron spectra for simple polyatomic molecules HCN, C₂H₂, and AH₂ type of hydrides can also be accounted for in terms of the slopes of the orbital energy curves in the correlation diagrams with respect to angles, as well as distances. Among all correlation diagrams, the slopes in the distance correlation diagram are related to the criterion for bond type—the positive for “bonding,” the negative for “antibonding,” while slopes with small magnitudes for “nonbonding.” The Fock matrix elements within the bond orbital basis provide heuristic and systematic rationalization of the slopes for the orbital energy curves. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 53–65, 2001     

Applicability of nondegenerate many-body perturbation theory to quasi-degenerate electronic states. II. A two-state model

In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four-electron models, in which the degree of quasi-degeneracy can be continuously varied, by the finite-order nondegenerate many-body perturbation theory, we examine in more detail a simple two-state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Padè approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi-reference perturbation approaches, which Kaldor advocates, are pointed out.     

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to ? = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for ? ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbó-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = ?14.6673473 corrected with these two truncation energy errors yields E = ?14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = ?14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.     

Perturbation theory for open-shell systems: Simulation of the UHF states by means of the perturbed RHF wave functions

Second-, third-, and selected fourth-order contributions to the correlation energy were calculated for a series of simple open-shell systems by means of the previously developed double-perturbation theory in the restricted MO formalism. It was found possible to assign some of the diagrams to self-consistency effects and to approximate in this way the E – E energy difference. A comparison is made with a more rigorous approach, in which the UHF ground-state wave function is expressed as a first-order perturbation expansion based on the RHF reference wave function. Distinguishing between “self-consistency” and “correlation” diagrams for open-shell systems in the RHF formulation represents a special case of a more general problem met in any double-perturbation treatment, such as, e.g., treatments of systems in the external field or perturbation expansions with noncanonical orbitals.     

Free energy derivatives: A new method for probing the convergence problem in free energy calculations

The convergence behavior of free energy calculations has been explored in more detail than in any previously reported work, using a model system of two neon atoms in a periodic box of water. We find that for thermodynamic integration-type free energy calculations as much as a nanosecond or more molecular dynamics sampling is required to obtain a fully converged value for a single λ point of the integrand. The concept of “free energy derivatives” with respect to the individual parameters of the force field is introduced. This formalism allows the total convergence of the simulation to be deconvoluted into components. A determination of the statistical “sampling ratio” from these simulations indicates that for window-type free energy calculations carried out in a periodic waterbox of typical size at least 0.6 ps of sampling should be performed at each window (0.7 ps if constraint contributions to the free energy are being determined). General methods to estimate and reduce the error in thermodynamic integration and free energy perturbation calculations are discussed. We show that the difficulty in applying such methods is determining a reliable estimate of the correlation length from a short series of data. © 1994 by John Wiley & Sons, Inc.     

The externally contracted CI method applied to N2

An approximate multireference CI method is presented. By grouping together configurations with the same internal parts and freezing their relative weights by the use of perturbation theory, the number of variational parameters is drastically reduced. The loss of correlation energy is shown to be usually less than 2%, and the timing is less than one ordinary CI iteration. Examples from calculations on some states of the nitrogen atom and nitrogen molecule are given. The basis set convergence for the lowest excitation energy in the atom is very slow. Less than 50% of the correlation effect is obtained at the s, p, d limit. After the inclusion of ? functions this value is improved to 83%. The dissociation energies of the molecule also show slow basis set convergence with errors of 0.5 eV even after addition of ? functions. The bond distances are, howeever, accurately reproduced with errors of less than 0.005 Å for all the states. A qualitative discussion of predissociation in the a ¹Π_g and B ³Π_gstates caused by spin–orbit interaction with the ⁵Σ_g⁺ state, is finally presented. Rapidly oscillating lifetimes between the different vibrational states are predicted.     

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Padé resummation methods are applied to high‐order series from multireference many‐body perturbation theory (MR‐MBPT) calculations using various partitioning schemes (Møller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low‐lying excited states of (i) CH₂, (ii) BeH₂ at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Padé approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Padé method generally outperforms the simple Padé resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Electron correlation and the compton profile of atomic neon

The radial momentum distribution I_o(p) and the Compton profile J_o(q) are determined for atomic neon from several restrictid Hartree-Fock (RHF) wavefunctions and two configuration interaction (CI) wavefunctions. The CI functions are the well correlated (full“second-order”) function of Viers, Schaeffer and Harris, and the Ahlrichs-Hinze multi-configuration Hartree-Fock (MCHF) function which includes only L-shell correlation. It is found for this completely closed shell system that the effects of electron correlation are quite small. This contrasts with the results for systems such as Be(²S) and B(²P) where the semi-internal and internal correlation effects were responsible for significant discrepancies between the RHF and CI results. These results indicate that a wavefunction which carefully includes the semi-internal, orbital polarization, and internal correlations beyond the RHF wavefunction (i.e., a “first-order” or “charge-density” function), should account for the principal correlation effects on the Compton profiles and momentum distributions.     

Generalized-molecular-orbital theory: Simple multiconfiguration self-consistent-field method

Even after completing a multiconfiguration self-consistent-field (MCSCF ) calculation, one must often include additional configuration interaction (CI ) to obtain quantitative or semiquantitative results. There is some question of whether the prior MCSCF calculation is worthwhile, if additional CI is needed later. We have developed a new MCSCF computational method, which, because of our assumptions about the nature of the configurations, yields one Fock-like operator for all the “filled” orbitals (high occupation numbers) and a second Fock-like operator for all the “virtual” orbitals (low occupation numbers). Since there are only two matrices to build, our method is considerably faster than other MCSCF approaches. Because of these similarities to standard molecular-orbital (MO ) calculations, we have termed our approach generalized-molecular-orbital (GMO ) theory. However, the “virtual” orbitals, unlike those of standard MO theory, are optimized to correlate the “filled” ones and can he used in a subsequent CI calculation. Results are presented for the correlation energy of H₂O, the spectroscopic constants of N₂, the singlet–triplet energy separations in CH₂, and the nature of the chromium–chromium quadruple bond. Although these results are at a very low level of CI , the GMO approach appears to correct for the gross deficiencies of the single-determinant SCF procedure.     

The effects of higher orders of perturbation theory on the correlation energy of atoms and bonds in molecules

We examine, for the first time, the effects of higher orders of Møller–Plesset perturbation theory on the individual atoms within a molecule and the bonds between them, via the topological energy partitioning method of interacting quantum atoms. In real terms (i.e., not by absolute value) MP3 decreases the correlation energy of a bond, and MP4SDQ also decreases the energy of the atoms at either end of the bond. In addition, we investigated long‐range through‐space dispersive effects on a H₂ oligomer. Overall, MP3 is the largest correction to the correlation energy, and most of that energy is allocated to chemical bonds, reducing their values in actual terms. The MP4SDQ bond correlation correction, despite being relatively small, tends to have two effects: (i) for small or negative correlation energies MP4SDQ tends to decrease the bond correlation values even more, and (ii) for large (positive) bond correlation energies MP4SDQ tends to restore the bond correlation energies from the MP3 back toward the MP2 values. Furthermore, each individual part of a molecule or complex (atom or bond) has a specific convergence pattern for the MPn series: through‐space interactions converge at MP2 but bonds converge at MP3 level. The atomic correlation energy appears to head toward convergence at the MP4 level.     

SCF and localized orbitals in ethylene: MBPT/CC results and comparison with one-million configuration Cl

Many-body perturbation theory (MBPT) and coupled-cluster (CC) calculations are performed on the ethylene molecule employing canonical SCF and simple bond-orbital localized orbitals (LO). Full fourth-order MBPT [i.e. SDTQ MBPT(4)], CC doubles (CCD) and CC singles and doubles (CCSD) energies are compared with the over one-million configuration ‘bench-mark” Cl calculation of Saxe et al. Though the SCF and LO reference determinant energies differ by 0.29706 hartree, the CCSD energy difference is only 1.7 mhartrees (mh). Our most extensive SCF orbital calculation, CCSD plus fourth-order triples, is found to be lower in energy than the CI result by 5.3 mh.     

Studies in perturbation theory XIII. Treatment of constants of motion in resolvent method,partitioning technique,and perturbation theory

After a brief survey of some basic concepts in the theory of linear spaces, the eigenvalue problem is formulated in the resolvent technique based on the introduction of a reference function φ and a complex variable ?. This leads to a series of fundamental concepts including the trial wave function, the inhomogeneous equation, and finally the transition and expectation values of the Hamiltonian, of which the former renders a “bracketing function” for the energy. In order to avoid the explicit limiting procedures in this approach, the eigenvalue problem is then reformulated in terms of the partitioning technique which, in turn, leads to a closed form of infinite-order perturbation theory. The eigenvalue problem is greatly simplified if the Hamiltonian H has a constant of motion Λ or has symmetry properties characterized by the group G = {g}, and the question is now how these simplifications can be incorporated into the partitioning technique and into perturbation theory. In both cases, there exists a set of projection operators {Q_k} which lead to a splitting of the Hilbert space into subspaces which have virtually nothing to do with each other. It is shown that, in the partitioning technique, it is sufficient to consider one of these subspaces at a time, and the results are then generalized to perturbation theory. It turns out that the finite-order expansions are no longer unique, and the commutation rules connecting the various forms are derived. The infinite-order results are finally presented in such a form that they are later suitable for the evaluation of upper and lower bounds to the energy eigenvalues.     

Simultaneous analytical optimization of variational parameters in Gaussian-type functions with full configuration interaction of multicomponent molecular orbital method by elimination of translational and rotational motions: application to isotopomers of the hydrogen molecule

We have extended the multicomponent molecular orbital (MCMO) method to the full-configuration interaction (full-CI) fully variational molecular orbital method by elimination of translational and rotational motion components from total Hamiltonian. In the MCMO scheme, the quantum effects of protons and deuterons as well as electrons can be directly taken into account. All variational parameters in the full-CI scheme, i.e., exponents and centers (alpha and R) in the Gaussian-type function (GTF) basis set as well as the CI coefficients, are simultaneously optimized by using their analytical gradients. The total energy of the H(2) molecule calculated using the electronic [6s3p2d1f] and nuclear [1s1p1d1f] GTFs is -1.161 726 hartree, which can be compared to the energy of -1.164 025 hartree reported using a 512 term-explicitly correlated GTF calculation. Although the d- and f-type nuclear GTFs contribute to the improvement of energy convergence, the convergence of electron-nucleus correlation energy is slower than that of electron-electron one. The nuclear wave functions are delocalized due to the electron-nucleus correlation effect compared to the result of Hartree-Fock level of MCMO method. In addition, the average internuclear distances of all diatomic molecules are within 0.001 A of the previously reported experimental results. The dipole moment of the HD molecule estimated by our method is 8.4 x 10(-4) D, which is in excellent agreement with the experimental result of (8-10) x 10(-4) D.     

精确固定节面量子Monte Carlo差值法

提出了精确固定节面量子Monte Carlo差值法, 这个新算法能够在精确固定节面量子Monte Carlo方法的基础上直接计算两个体系之间的能量差, 且使计算结果的统计误差达到10-5 hartree 数量级, 获得电子相关能90%以上. 我们把这个新算法应用于分子势能面的研究中, 使用一个“刚性移动”模型, 利用Jacobi变换使分子两个几何构型的能量计算具有很好的正相关性, 因而能得到准确的能量差值, 由此就可以得到精确的分子势能面.     

Application of the many-body perturbation theory by using localized orbitals

Diagrammatic formulation of the many-body perturbation theory is investigated when both the occupied orbitals and the virtual ones are localized, i.e., they are unitary transforms of the canonical Hartree–Fock orbitals. All diagrams representing ground state correlation energy can be generated through fifth order. For cyclic polyenes C₆H₆ and C₁₀H₁₀ as model systems, the energy corrections are calculated in the Pariser–Parr–Pople approximation for a wide range of the coupling constant β^?1, through fourth order including some fifth order terms. The results are compared to those obtained by other methods: perturbation theory by using canonical orbitals and full CI. The effect of neglecting contributions from orbitals localized into neighboring sites is also studied.