Third-order multireference perturbation theory: the n-electron valence state perturbation-theory approach

A formulation of the n-electron valence state perturbation theory (NEVPT) at the third order of perturbation is presented. The present implementation concerns the so-called strongly contracted variant of NEVPT, where only a subspace of the first-order interacting space is taken into account. The resulting strongly contracted NEVPT3 approach is discussed in three test cases: (a) the energy difference between the 3B1 and 1A1 states of the methylene molecule, (b) the potential-energy curve of the N2 molecule ground state, and (c) the chromium dimer (Cr2) ground-state potential-energy profile. Particular attention is devoted to the last case where large basis sets comprising also h orbitals are adopted and where remarkable differences between the second- and third-order results show up.     

New perspectives in multireference perturbation theory: the n-electron valence state approach

The n-electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory which is based on a zero order reference wavefunction of CAS-CI type (complete active space configuration interaction) and which is characterized by the utilization of correction functions (zero order wavefunctions external to the CAS) of multireference nature, obtained through the diagonalization of a suitable two-electron model Hamiltonian (Dyall’s Hamiltonian) in some well defined determinant spaces. A review of the NEVPT approach is presented, starting from the original second order state-specific formulation, going through the quasidegenerate multi-state extension and arriving at the recent implementations of the third order in the energy and of the internally contracted configuration interaction. The chief properties of NEVPT—size consistence and absence of intruder states—are analyzed. Finally, an application concerning the calculation of the vertical spectrum of the biologically important free base porphin molecule, is presented.     

An application of second-order n-electron valence state perturbation theory to the calculation of excited states

n–electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory where all the zero-order wave functions are of multireference nature, being generated as eigenfunctions of a two–electron model Hamiltonian. The absence of intruder states makes NEVPT an interesting choice for the calculation of electronically excited states. Test calculations have been performed on several valence and Rydberg transitions for the formaldehyde and acetone molecules; the results are in good accordance with the best calculations and with the existing experimental data.Contribution to the Jacopo Tomasi Honorary Issue     

Ab initio n-electron valence state perturbation theory study of the adiabatic transitions in carbonyl molecules: formaldehyde, acetaldehyde, and acetone

The application of the recently developed second-order n-electron valence state perturbation theory (NEVPT2) to small carbonyl molecules (formaldehyde, acetaldehyde, and acetone) is presented. The adiabatic transition energies are computed for the singlet and triplet n-->pi(*), pi-->pi(*), and sigma-->pi(*) states performing a full geometry optimization of the relevant states at the single state CASSCF level and taking into account the zero point energy correction in the harmonic approximation. The agreement with the known experimental values and with previously published high level calculations confirms that NEVPT2 is an efficient tool to be used for the interpretation of molecular electronic spectra. Moreover, different insight into the nature of the excited states has been obtained. Some of the transitions presented here have never been theoretically computed previously [(3)(pi-->pi(*)) and (3)(sigma-->pi(*)) adiabatic transitions in acetaldehyde and acetone] or have been studied only using moderate level (single reference based) ab initio methods (all adiabatic transitions in acetaldehyde). In the present work a consistent disagreement between NEVPT2 and experiment has been found for the (3)(pi-->pi(*)) adiabatic transition in all molecules: this result is attributed to the low intensity of the transition to the first vibrational levels of the excited state. The n-->pi(*) singlet and triplet vertical transition energies are also reported for all the molecules.     

Generalized perturbation theory of effective valence shell hamiltonians

The second quantized effective valence shell hamiltonian of Iwata and Freed is generalized to incorporate valence orbital energies into the perturbative (matrix) energy denominators, eliminating convergence difficulties in calculations of atomic valence shell hamiltonians. When the matrix energy denominators are taken to be the simplest form our generalized effective hamiltonian reduces to Brandow quasi-degenerate theory.     

Paired-permanent approach to valence bond theory

A new function called paired-permanent is defined and widely discussed, and a practicable procedure for evaluations of paired-permanents is proposed, which is similar to the Laplace method for determinants. Using the concept of paired-permanents, an efficient algorithm is presented for evaluating the Hamiltonian and overlap matrix elements in the spin-free form of valence bond (VB) theory. With the new algorithm, a spin-free wavefunction is simply written as a paired-permanent, and an overlap matrix element may be obtained by evaluating a corresponding paired-permanent. Meanwhile, the Hamiltonian matrix element is expressed in terms of the summation of the products of electronic integrals and the corresponding sub-paired-permanents     

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.     

On the concept of “Partitions” in the perturbation theory of n-electron systems

This paper deals with the perturbation theory of an n-electron Hamiltonian of the general form H = ∑ⁿ ?(i) + λ∑ⁿ g(i, j) = H (f, g; n). In comparison to the Brueckner–Goldstone diagrammatic perturbation theory, we adopt the more general standpoint of admitting, for the construction of an n-particle state, component states of 1, 2, 3, and more particles [O. Sinanoglu, Phys. Rev. 122 , 493 (1961) and C. D. H. Chisholm and A. Dalgarno, Proc. R. Soc. (London) Sec. A 292 , 264 (1966)]. We show that this leads to the concept of a “partition” of a perturbational eigenstate (or energy) of H. A “partition” is a natural decomposition which: (i) is finite; (ii) relates the eigenvalue problem of the system H = H (f, g; n) to those of certain subsystems H (f, g; n₁)(n₁ < n); (iii) uses “nonseparable” components. We domonstrate (under the preliminary assumption of “strict” nondegeneracy) the second-order energy to possess a “partition.” The components therein are second-order energies of two- and three-particle states. The proof uses an extension of Racah's concept of the fractional-parentage expansion.     

Equation of state based on the weeks-chandler-andersen perturbation theory

An equation of state based on the Weeks-Chandler-Andersen (WCA) perturbation theory utilizing the simplified random-phase approximation (RPA) term is p of state is applied successfully to calculate the P-V-T relationship for a spherical molecule, argon.For nonspherical molecules, the effects of anisotropic interactions are treated empirically. The calculated P-V-T relationships and saturated properties for nonspherical and nonpolar molecules agree well with experimental data. The potential parameters for nonpolar substances are well correlated with the acentric factors.     

Application of SCF perturbation theory to the study of tetrahedrally bonded valence crystals

The bond interaction method has been applied to the study of hexagonal and cubic BeO, BN and diamond. It is found that the CNDO method yields the cubic structure as the more stable forms of BN and diamond; the BeO calculation did not converge well. It is concluded that the method is limited to the study of relatively covalent systems.     

The V state of ethylene: valence bond theory takes up the challenge

The ground state and first singlet excited state of ethylene, so-called N and V states, respectively, are studied by means of modern valence bond methods. It is found that extremely compact wave functions, made of three VB structures for the N state and four structures for the V state, provide an N → V transition energy of 8.01 eV, in good agreement with experiment (7.88 eV for the N → V transition energy estimated from experiments). Further improvement to 7.96/7.93 eV is achieved at the variational and diffusion Monte Carlo (MC) levels, respectively, VMC/DMC, using a Jastrow factor coupled with the same compact VB wave function. Furthermore, the measure of the spatial extension of the V state wave function, 19.14 a ₀ ² , is in the range of accepted values obtained by large-scale state-of-the-art molecular orbital-based methods. The σ response to the fluctuations of the π electrons in the V state, known to be a crucial feature of the V state, is taken into account using the breathing orbital valence bond method, which allows the VB structures to have different sets of orbitals. Further valence bond calculations in a larger space of configurations, involving explicit participation of the σ response, with 9 VB structures for the N state and 14 for the V state, confirm the results of the minimal structure set, yielding an N → V transition energy of 7.97 eV and a spatial extension of 19.16 a ₀ ² for the V state. Both types of valence bond calculations show that the V state of ethylene is not fully ionic as usually assumed, but involving also a symmetry-adapted combination of VB structures each with asymmetric covalent π bonds. The latter VB structures have cumulated weights of 18–26 % and stabilize the V state by about 0.9 eV. It is further shown that these latter VB structures, rather than the commonly considered zwitterionic ones, are the ones responsible for the spatial extension of the V state, known to be ca. 50 % larger than the V state.     

Third-order many-body perturbation theory for the ground state of the carbon monoxide molecule

Many-body perturbation calculations have been performed for the ground state of the carbon monoxide molecule at its equilibrium internuclear separation. The calculations are complete through third order within the algebraic approximation; i.e., the state functions are parameterized by expansion in a finite basis set. All two-, three-, and four-body terms are rigorously determined, and many-body effects are found to be very important. A detailed comparison is made with a previously reported configuration interaction study. Padé approximants to the energy expansion are constructed. The many-body perturbative wave function is used in the Rayleigh quotient to produce upper bounds to the electronic energy.     

A study of the ground state of manganese dimer using quasidegenerate perturbation theory

We study the electronic structure of the ground state of the manganese dimer using the state-averaged complete active space self-consistent field method, followed by second-order quasidegenerate perturbation theory. Overall potential energy curves are calculated for the 1Sigmag+, 11Sigmau+, and 11Piu states, which are candidates for the ground state. Of these states, the 1Sigmag+ state has the lowest energy and we therefore identify it as the ground state. We find values of 3.29 A, 0.14 eV, and 53.46 cm(-1) for the bond length, dissociation energy, and vibrational frequency, in good agreement with the observed values of 3.4 A, 0.1 eV, and 68.1 cm(-1) in rare-gas matrices. These values show that the manganese dimer is a van der Waals molecule with antiferromagnetic coupling.     

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.     

Plutonium valence state distributions

A method for ascertaining equilibrium valence state distributions of plutonium in acid solutions as a function of the plutonium oxidation number and the solution acidity is illustrated with an example. The method may be more practical for manual use than methods based upon polynomial equations. Mound Laboratory is operated by Monsanto Research Corporation for the U.S. Atomic Energy Commission under Contract No. AT-33-1-GEN-53.     

Applicability of quasi-degenerate many-body perturbation theory to the ground state of the F2 molecule

The ground-state potential energy curve of the F₂ molecule as well as spectroscopic constants were calculated by means of the second-order quasi-degenerate many-body perturbation theory within a full (eight) valence orbital space using a DZP basis set. The problem encountered with a large number of valence electrons is avoided by a proper redefinition of the Fermi vacuum. A comparison with other related multireference techniques is also provided. © 1995 John Wiley & Sons, Inc.     

Stationary perturbation theory

Summary After a short recapitulation of the basic concepts of stationary perturbation theory, this is applied to a many-electron Hamiltonian, with or without an external field, given in a Fock space formulation in terms of a finite basis, the exact eigenfunctions of which are the full-CI wave functions. The Lie algebra _c ⁿ of the variational group corresponding to this problem is presented. It has an important subalgebra _c ⁽¹⁾ of one-particle transformations. Hartree-Fock and coupled Hartree-Fock (also uncoupled Hartree-Fock) as well as MC-SCF and coupled MC-SCF are outlined in this framework. Many-body perturbation theory and Møller-Plesset perturbation theory are derived from the same kind of stationarity condition and a new non-perturbative iteration construction of the full-CI wave function is proposed, the first Newton-Raphson iteration cycle of which is CEPA-0. For the treatment of electron correlation for properties two variants of Møller-Plesset theory referred to as coupled (CMP) and uncoupled (UCMP) are defined, neither of which is fully satisfactory. While CMP satisfies a Brillouin condition, which implies that first order correlation corrections to first- and second-order properties vanish, it does not satisfy a Hellmann-Feynman theorem, i.e. a first order property isnot the expectation value of the operator associated with the property. Conversely UCMP satisfies a Hellmann-Feynman theorem but no Brillouin theorem. The incompatibility of the two theorems is related to an unbalanced treatment of one-particle- and higher excitations in MP theory. CMP, which is based on coupled Hartree-Fock as uncorrelated reference, appears to have slight advantages over UCMP, but neither variant looks very promising for the evaluation of 2nd order correlation corrections to 2nd-order properties. Then four variants of the perturbation theory of properties with a nonperturbative treatment of electron correlation on CEPA-0 level (but extendable to a higher level) are discussed. While those variants which are the direct counterpart of UCMP and CMP must be discarded, the perturbative CEPA-0 derived from a perturbative treatment on full-CI level appears to satisfy all important criteria, in particular it satisfies a Brillouin-Brueckner condition and a Hellmann-Feynman theorem. A simplified version, the coupled Brillouin-Brueckner CEPA-0 appears to have essentially the same qualities. It is important to replace the Brillouin condition of MP theory by the Brillouin-Brueckner condition in non-perturbative approaches, especially if one is interested in properties.     

Large-Order perturbation theory

The original motivation for studying the asymptotic behavior of the coefficients of perturbation series came from quantum field theory. An overview is given of some of the attempts to understand quantum field theory beyond finite-order perturbation series. At least in the case of the Thirring model and probably in general, the full content of a relativistic quantum field theory cannot be recovered from its perturbation series. This difficulty, however, does not occur in quantum mechanics, and the anharmonic oscillator is used to illustrate the methods used in large-order perturbation theory. Two completely different methods are discussed, the first one using the WKB approximation, and a second one involving the statistical analysis of Feynman diagrams. The first one is well developed and gives detailed information about the desired asymptotic behavior, while the second one is still in its infancy and gives instead information about the distribution of vertices of the Feynman diagrams.