Comparison of MBPT and coupled-cluster methods with full CI. Importance of triplet excitation and infinite summations

Results from full fourth-order perturbation theory [SDTQ MBPT(4)], and the coupled-cluster single- and double-excitation model (CCSD). are compared with recent full CI results for BH, HF, NH₃, and H₂O. For H₂O, studies include large symmetric displacements of the OH bonds, which offer a severe test for any single-reference MBPT/CC method. In every case. CCSD plus fourth-order triple-excitation terms provide agreement with the full CI to < 2 kcal/mole. SDTQ MBPT(4) has an error 10 kcal/mole for displaced H₂O.     

A note on the convergence of multiconfigurational many-body perturbation theory

The convergence of multiconfigurational many-body perturbation theory (MC MBPT ) is discussed in connection with the intruder state. Its convergence properties are first examined with a fictitious three-level system employing a Hermitian version of MC MBPT , which permits a general model space. It is then applied to the H₂—H₂ and N₂ systems. The results suggest that a more extensive model space is likely to embrace new intruder states and the space extension be executed with due caution.     

Parallel implementation of electronic structure energy, gradient, and Hessian calculations

ACES III is a newly written program in which the computationally demanding components of the computational chemistry code ACES II [J. F. Stanton et al., Int. J. Quantum Chem. 526, 879 (1992); [ACES II program system, University of Florida, 1994] have been redesigned and implemented in parallel. The high-level algorithms include Hartree-Fock (HF) self-consistent field (SCF), second-order many-body perturbation theory [MBPT(2)] energy, gradient, and Hessian, and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] energy and gradient. For SCF, MBPT(2), and CCSD(T), both restricted HF and unrestricted HF reference wave functions are available. For MBPT(2) gradients and Hessians, a restricted open-shell HF reference is also supported. The methods are programed in a special language designed for the parallelization project. The language is called super instruction assembly language (SIAL). The design uses an extreme form of object-oriented programing. All compute intensive operations, such as tensor contractions and diagonalizations, all communication operations, and all input-output operations are handled by a parallel program written in C and FORTRAN 77. This parallel program, called the super instruction processor (SIP), interprets and executes the SIAL program. By separating the algorithmic complexity (in SIAL) from the complexities of execution on computer hardware (in SIP), a software system is created that allows for very effective optimization and tuning on different hardware architectures with quite manageable effort.     

Application of many-body perturbation theory in the localized representation for the all-trans conjugated polyenes

Ab initio self-consistent field (SCF ) calculations are performed with the standard 6-31G* basis set for all-trans conjugated oligomers C_2n+2H_2n+4. The canonical occupied and virtual molecular orbitals (MO s) are separately localized by unitary transformations. Due to the localization, the perturbation operator is partitioned into two-particle and into single-particle terms; the MBPT is, therefore, a double-perturbation expansion in this case. By using the localized representation of the MBPT , the correlation energy contributions can be partitioned into local and nonlocal effects. It can be shown that the local effects are very important and well transferable, which makes possible the calculation of the correlation energy of larger molecules if the localized molecular orbitals (occupied and virtual) of smaller related molecules are known. © 1994 John Wiley & Sons, Inc.     

Information theoretic indices for characterization of chemical structures. By Danail Bonchev Wiley,New York, 1983

Wilson, Jankowski, and Paldus have recently applied nondegenerate many-body perturbation theory (MBPT ) to simple models, in which the degree of quasidegeneracy could be varied continuously, and concluded that the nondegenerate theory was applicable even near degeneracy. The error in their results changes, however, considerably with geometry, leading to an incorrect potential surface. An extension of their calculations shows convergence even at exact degeneracy (square planar H₄). It is shown here that the apparently good convergence is due to the suppression of the large (infinite at exact degeneracy) component of the perturbation energy in low order by the way the Hamiltonian is partitioned. This component will, however, resurface at higher orders, leading to slow convergence or even divergence. The low-order sum of the perturbation series is not very meaningful, depends strongly on details of the zero-order Hamiltonian, and yields, in general, incorrect potential surfaces. Multireference MBPT eliminates these problems.     

Direct versus indirect many-body methods for calculating vertical electron affinities: applications to F−, OH− , NH2−, CN−, Cl−, SH− and PH2−

Electron propagator theory (EPT) is applied to calculating vertical ionization energies of the anions F⁻, Cl⁻, OH⁻,SH⁻, NH₂⁻, PH₂⁻ and CN⁻. Third-order and outer valence approximation (OVA) quasiparticle calculations are compared with ΔMBPT(4) (MBPT, many-body perturbation theory) results using the same basis sets. Agreement with experiment is satisfactory for EPT calculations except for F⁻ and OH⁻, while the ΔMBPT treatments fail for CN⁻. EPT(OVA) estimates are reliable when the discrepancy between second- and third-order results is small. Computational aspects are discussed, showing relative merits of direct and indirect methods for evaluating electron binding energies.     

Use of coupled-cluster based linear response theory and multireference hermitian MBPT to IP calculations of HF

We report in this paper the results of outer and inner valence IP calculations for the HF molecule using two different many-body methods for the direct evaluation of energy differences. The first is the nonperturbative coupled-cluster based linear response theory (LRT) and the second is the hermitian open-shell many-body perturbation theory (MBPT). A Huzinaga-Dunning (9s5p→ 5s3p/3s) basis has been used. LRT uses an “ionization operator” S as in the equation of motion method (EOM) to generate the ionized states from a coupled-cluster type of ground state. S is chosen to consist of single ionization and ionization-cum-shake-up operators, thus treating the Koopmans as well as the shake-up states on equal footing. LRT would thus be capable of computing both the outer and the inner valence regions with equal facility. This is borne out by the results. For the open-shell MBPT, the model space is chosen to be spanned by the singly ionized determinants. The convergence of the results for the inner valence region is slow, and the results obtained from the [2, 1] Pade' approximants are presented. Unlike the LRT, the inner valence region is not reproduced with full complexity in MBPT, indicating that it is essential to modify the theory by way of expanding the model space to contain the shake-up determinants also.     

Ab initio density functional theory applied to quasidegenerate problems

Ab initio density functional theory (DFT), previously applied primarily at the second-order many-body perturbation theory (MBPT) level, is generalized to selected infinite-order effects by using a new coupled-cluster perturbation theory (CCPT). This is accomplished by redefining the unperturbed Hamiltonian in ab initio DFT to correspond to the CCPT2 orbital dependent functional. These methods are applied to the Be-isoelectronic systems as an example of a quasidegenerate system. The CCPT2 variant shows better convergence to the exact quantum Monte Carlo correlation potential for Be than any prior attempt. When using MBPT2, the semicanonical choice of unperturbed Hamiltonian, plays a critical role in determining the quality of the obtained correlation potentials and obtaining convergence, while the usual Kohn-Sham choice invariably diverges. However, without the additional infinite-order effects, introduced by CCPT2, the final potentials and energies are not sufficiently accurate. The issue of the effects of the single excitations on the divergence in ordinary OEP2 is addressed, and it is shown that, whereas their individual values are small, their infinite-order summation is essential to the good convergence of ab initio DFT.     

Applications of the MBPT in the localized representation

Diagrammatic formulation of the MBPT is applied when the occupied and the virtual canonical orbitals are separately localized by unitary transformations. In this localized representation, due to the off-diagonal Fock matrix elements, the perturbation operator contains extra terms generating the so-called localization corrections. These corrections enter the perturbation energy in third and higher orders. Their magnitude depends on the type of localization, but they represent only a small fraction of the canonical corrections. The calculation of the localization corrections, however, does not need a significant amount of extra computer time. It is shown that by introducing an “order of neighborhood” local and nonlocal effects of the electron correlation can be separated and the contribution of the nonlocal effects can be neglected to a good approximation. Ab initio calculations have been carried out for the normal saturated hydrocarbons: C_2n+1H_4n+4 and for the all-trans conjugated polyenes C_2n+2H_2n+4. As to the ratio of the local and nonlocal corrections, it is shown that there is only a quantitative difference for these two kinds of systems (strongly or weakly localizable). Neglecting nonlocal effects, considerable amount of computer time can be saved.     

CCSD(T) expectation value calculations of first-order properties

An expectation value approach to calculations of first-order properties using the non-iterative, triple-excitation amplitudes in the coupled cluster wave function is exploited. Three methods are suggested and analysed using the many body perturbation theory (MBPT) expansion arguments. The first method, in which non-iterative triple-excitation amplitudes are used in the expression for the expectation values, makes the wave function accurate through the second order of MBPT. In the second method, which is an extension of the first, effects of triple-excitation amplitudes are coupled with single- and double-excitation amplitudes. The correlated density matrix equivalent through the fourth order to that obtained when CCSDT-la amplitudes are used is employed in the third method. The suggested methods are tested on dipole moment and polarizability calculations for several diatomic closed-shell molecules and are compared to other related approaches. Received: 15 May 1997 / Accepted: 5 June 1997     

A comparison of different many-body perturbation theory calculations of the ground state of SiS

A comparison of different many-body perturbation theory (MBPT ) calculations of the ground state rotational and vibrational constants of SiS is made. The calculations are performed up to the complete fourth-order MBPT level, and in all cases two basis sets are utilized. The results of the third-order and some incomplete fourth-order calculations are in good agreement, but the complete fourth-order is among the worst as compared with the experimental data. Analysis of the different contributions to the calculated correlation eneriges points towards the necessity of including even higher-order terms of the(MBPT ) expansion.     

A fast CI method for closed shells

A variational method for closed shells derived from MBPT is proposed. The method consists in the diagonalization of the hamiltonian in the subspace generated by the HF ground state |φ_HF) and f_i(H_o) V_res|φ_HF) (i = 1,2,… m) where H_o is the HF hamiltonian, V_res is the residual potential and the f_iti are arbitrary functions. Test calculations performed on H₂O give 97% of the double excitations CI limit with very short computer time.     

Localized orbitals for the description of molecular interaction

The intermolecular interaction between the molecules CH₂O and NH₃ was investigated by the supermolecule method. The interaction energies were first calculated at the ab initio SCF level, and the electron correlation was included via second-order Møller-Plesset perturbation theory (MP 2). The basis set superposition error (BSSE ) was taken into account by the counter-poise (CP ) method. The occupied and the virtual canonical molecular orbitals (CMOS ) of the supermolecule were separately localized by the Boys' procedure. The correlation correction was calculated by the many-body perturbation theory (MBPT ) in the localized representation. Contributions of the third- and fourth-order localized diagrams were added to those of the second-order canonical diagram. This procedure gives a correction nearly equivalent to that of MP 2. The possibility to separate LMO contributions responsible for the dispersion interaction was investigated.     

A quantum chemical study of the hydrogen bonding in the CO2⋯HF and N2O⋯HF complexes

Summary Quantum chemical ab initio calculations have been performed for the complex CO₂HF and N₂OHF. The interaction energies were computed through fourth order MBPT and were corrected for basis set superposition errors. Extended polarized basis sets were used which are constructed to give accurate values for electric moments and polarizabilities. The complex NNOHF was found to be bent, while OCOHF is linear, in agreement with experiment. The MBPT calculations give evidence for a second linear isomeric structure FHNNO, a possibility which has also been suggested by recent experimental data. The computed binding energies are: 2.5 kcal/mol for OCOHF, 2.4 kcal/mol for NNOHF, and 3.0 kcal/mol for FHNNO. At the SCF level, the FHNNO complex is less stable than NNOHF, but correlation has a large effect on the geometry and energetics of the latter complex. The NNOHF complex seems to be a system where the positive intramolecular correlation correction prevails over the negative intermolecular component.     

Many-body perturbation theory,coupled-pair many-electron theory,and the importance of quadruple excitations for the correlation problem

Many-body (diagrammatic) perturbation theory (MBPT ), coupled-pair many-electron theory (CPMET ), and configuration interaction (CI ) are investigated with particular emphasis on the importance of quadruple excitations in correlation theories. These different methods are used to obtain single, double, and quadruple excitation contributions to the correlation energy for a series of molecules including CO₂, HCN, N₂, CO, BH₃, and NH₃. It is demonstrated that the sum of double and quadruple excitation diagrams through fourth-order perturbation theory is usually quite close to the CPMET result for these molecules at equilibrium geometries. The superior reliability of the CPMET model as a function of internuclear separation is illustrated by studying the ¹∑ potential curve of Be₂. This molecule violates the assumption common to nondegenerate perturbation theory that only a single reference function is important and this causes improper behavior of the potential curve as a function of R. This is resolved once the quadruple excitation terms are fully included by CPMET .     

Fourth-order diagrammatic MB–RSPT calculations of the correlation energy: N2, CO,F2 and the reaction energy of the process ½F2 + ½H2 = HF

The correlation energies obtained by the fourth-order diagrammatic perturbation theory were analyzed for three diatomic molecules: N₂, CO, and F₂. The results were compared with correlation energies obtained previously for the ten-electron hydrides HF, H₂O, and NH₃. The relative importance of contributions which arise from the double excitations, from the quadruple excitations, as well as from the renormalization term was investigated. It is shown that for the diatomic molecules under study these contributions are considerably larger than for the ten-electron hydrides not only in absolute value but also in percentage: they represent about 3, 3, and 5%, respectively, of the valence shell correlation energy obtained by the perturbation theory up to the fourth order. A careful analysis of the fourth-order correlation effects is also presented for the reaction energy of the process ½H₂ + ½F₂ = HF.     

Electron correlation in weakly coupled transition metal dimers: Bimetallocenylenes and bimetallocenes

The Hartree-Fock (HF) instabilities in a series of bimetallocenes (1) and bimetallocenylenes (2) with Fe, Co, Ni and Cr as 3d centers have been investigated by means of a semiempirical INDO Hamiltonian. The HF picture is only valid in the case of the iron dimers. Strong correlation effects are encountered in the Co, Ni and Cr complexes. The necessary conditions for singlet, non-singlet (triplet) and non-real variations of the HF orbitals are discussed in detail. Singlet fluctuations are the result of intraatomic angular correlation (short-range) at each 3d center. The violation of the spin symmetry corresponds to a long-range interaction between the transition metal centers. Only for MOs with large 3d _xz amplitudes there exists a channel for the interatomic spin decoupling. Consequences for polymetallocenes are shortly discussed.     

Beryllium dimer,a critical test case of MBPT and CI methods

The ground-state potential curve for the beryllium dimer is calculated as a critical test case for methods based on many-body perturbation theory (MBPT ) and configuration interaction (CI ). In particular, the recently proposed double excitation (DE ) MBPT method is compared to the standard SCF-CI method including single and double excitations from a single reference determinant. The SCF-CI method is shown to give surprisingly accurate results compared to more complete CI calculations including a larger configuration space, whereas the DE-MBPT method breaks down more or less completely, particularly for larger basis sets. The results thus demonstrate the importance of including the renormalization terms in this case. Finally, Davidson's correction and related methods lead to an even more severe breakdown than the DE-MBPT method.     

Finite-field many-body perturbation theory IV. Basis set optimization in MBPT calculations of molecular properties. Molecular quadrupole moments

The choice of truncated basis sets and their optimization for MBPT calculations of molecular properties are discussed. It is pointed out that computing the correlation corrections to some kth order property by using the MBPT approach requires the knowledge of accurate perturbed orbitals through the kth order. Hence, it is argued that the basis set functions can be optimized with respect to the perturbed energies calculated within the coupled Hartree-Fock method. The proposed procedure is illustrated by MBPT calculations of quadrupole moments of H₂ and FH. Additionally, also some estimates of the quadrupole polarizability tensor components for these molecules are obtained.     

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.