Third-order Energies Research Articles

Third-Order Møller-Plesset Perturbation Theory Made Useful? Choice of Orbitals and Scaling Greatly Improves Accuracy for Thermochemistry, Kinetics, and Intermolecular Interactions.

We develop and test methods that include second- and third-order perturbation theory (MP3) using orbitals obtained from regularized orbital-optimized second-order perturbation theory, κ-OOMP2, denoted as MP3:κ-OOMP2. Testing MP3:κ-OOMP2 shows RMS errors that are 1.7-5 times smaller than those of MP3 across 7 data sets. To do still better, empirical training of the scaling factors for the second- and third-order correlation energies and the regularization parameter on one of those data sets led to an unregularized scaled (c2 = 1.0; c3 = 0.8) denoted as MP2.8:κ-OOMP2. MP2.8:κ-OOMP2 yields significant additional improvement over MP3:κ-OOMP2 in 4 of 6 test data sets on thermochemistry, kinetics, and noncovalent interactions. Remarkably, these two methods outperform coupled cluster with singles and doubles in 5 of the 7 data sets considered, at greatly reduced cost (no iterations).

Inner Space Perturbation Theory in Matrix Product States: Replacing Expensive Iterative Diagonalization.

We propose an inner space perturbation theory (isPT) to replace the expensive iterative diagonalization in the standard density matrix renormalization group theory (DMRG). The retained reduced density matrix eigenstates are partitioned into the active and secondary space. The first-order wave function and the second- and third-order energies are easily computed by using one step Davidson iteration. Our formulation has several advantages including (i) keeping a balance between the efficiency and accuracy, (ii) capturing more entanglement with the same amount of computational time, (iii) recovery of the standard DMRG when all the basis states belong to the active space. Numerical examples for the polyacenes and periacene show that the efficiency gain is considerable and the accuracy loss due to the perturbation treatment is very small, when half of the total basis states belong to the active space. Moreover, the perturbation calculations converge in all our numerical examples.

Degenerate density perturbation theory

Fractional occupation numbers can be used in density functional theory to create a symmetric Kohn-Sham potential, resulting in orbitals with degenerate eigenvalues. We develop the corresponding perturbation theory and apply it to a system of $N_d$ degenerate electrons in a harmonic oscillator potential. The order-by-order expansions of both the fractional occupation numbers and unitary transformations within the degenerate subspace are determined by the requirement that a differentiable map exists connecting the initial and perturbed states. Using the X$\alpha$ exchange-correlation (XC) functional, we find an analytic solution for the first-order density and first through third-order energies as a function of $\alpha$, with and without a self-interaction correction. The fact that the XC Hessian is not positive definite plays an important role in the behavior of the occupation numbers.

General-Order Many-Body Green's Function Method.

Electron binding energies are evaluated as differences in total energy between the N- and (N ± 1)-electron systems calculated by the nth-order Møller-Plesset perturbation (MPn) theory using the same set of orbitals. The MPn energies up to n = 30 are, in turn, obtained by the determinant-based method of Knowles et al. (Chem. Phys. Lett. 1985, 113, 8-12). The zeroth- through third-order electron binding energies thus determined agree with those obtained by solving the Dyson equation in the diagonal and frequency-independent approximations of the self-energy. However, as n → ∞, they converge at the exact basis-set solutions from the Dyson equation with the exact self-energy, which is nondiagonal and frequency-dependent. This suggests that the MPn energy differences define an alternative diagrammatic expansion of Koopmans-like electron binding energies, which takes into account the perturbation corrections from the off-diagonal elements and frequency dependence of the irreducible self-energy. Our analysis shows that these corrections are included as semireducible and linked-disconnected diagrams, respectively, which are also found in a perturbation expansion of the electron binding energies of the equation-of-motion coupled-cluster methods. The rate of convergence of the electron binding energies with respect to n and its acceleration by Padé approximants are also discussed.

A new size extensive multireference perturbation theory

A new multireference perturbation series is derived based on the Rayleigh-Schrödinger perturbation theory. It is orbitally invariant. Its computational cost is comparable to the single reference Møller-Plesset perturbation theory. It is demonstrated numerically that the present multireference second- and third-order energies are size extensive by two types of supermolecules composed of H2 and BH monomers. Spectroscopic constants of F2(X1Σg+),Cl2(X1Σg+),C2-(X2Σg+),B2(X3Σg-),and C2+(X4Σg-) as well as the ground state energies of H2O, NH2, and CH2 at three bond lengths have been calculated with the second multireference perturbation theory. The dissociation behaviors of CH4 and HF have also been investigated. Comparisons with other approximate theoretical models as well as the experimental data have been carried out to show their relative performances.

All-order perturbation calculation of energies, hyperfine constants, multipole polarizabilities, and blackbody radiation shift inSr87+

Excitation energies of the $[\mathrm{Kr}]{\mathit{ns}}_{1/2}$, $[\mathrm{Kr}]{\mathit{np}}_{j}$, $[\mathrm{Kr}]{\mathit{nd}}_{j}$, and $[\mathrm{Kr}]{\mathit{nf}}_{j}$ ($n\ensuremath{\leqslant}9$ and $[\mathrm{Kr}]=(1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}$) in Sr ii are evaluated. First-order, second-order, third-order, and all-order Coulomb energies and first-order and second-order Coulomb-Breit energies are calculated. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the levels up to $n=7$. Electric-dipole ($5{s}_{1/2}\ensuremath{-}{\mathit{np}}_{j}$, $n=5--26$), electric-quadrupole ($5{s}_{1/2}\ensuremath{-}{\mathit{nd}}_{j}$, $n=4--26$), and electric-octupole ($5{s}_{1/2}\ensuremath{-}{\mathit{nf}}_{j}$, $n=4--26$) matrix elements are calculated to obtain the ground-state $E1$, $E2$, and $E3$ static polarizabilities. Scalar and tensor polarizabilities for the $5{p}_{j}\ensuremath{-}9{p}_{j}$ and $4{d}_{j}\ensuremath{-}8{d}_{j}$ excited states in Sr ii are also calculated. All the above-mentioned matrix elements are determined using the all-order method. We also investigate the hyperfine structure in $^{87}\mathrm{Sr}$${}^{+}$. The hyperfine $A$ values and $B$ values are determined for the first low-lying levels up to $n=7$. The quadratic Stark effect on hyperfine-structure levels of the $^{87}\mathrm{Sr}$${}^{+}$ ground state is investigated. The calculated shift for the $(F=5,M=0)\ensuremath{\leftrightarrow}(F=4,M=0)$ transition is found to be 0.120(1) Hz/(kV/cm)${}^{2}$. These calculations provide a theoretical benchmark for comparison with the experiment and theory. A careful study of uncertainty of our calculations is carried out for the transition-matrix elements, line strengths, transition rates, lifetimes, polarizabilities, and the Stark shift coefficient.

Long- and intermediate-range interaction between two ground state hydrogen atoms

Perturbation theory in the polarization approximation has been employed, without using the multipole expansion for the interaction Hamiltonian, to calculate the second- and third-order interaction energies of two ground state hydrogen atoms for internuclear separations R - 4(2) 12 a.u. Using recently computed variational energies for H-H and H-H, the values of higher order corrections to the interaction energy have been estimated. The accuracy of several approximations commonly used in calculations of inter-molecular interaction energies has been tested.

Third-order dispersion energy from response functions

In the second order of perturbation theory, the dispersion component of the interaction energy of two atoms or molecules can be expressed as an integral of frequency-dependent response functions of monomers. An analogous formula is derived here for the third-order dispersion energies. The general formula involves a nonstandard response function, but for atoms in asymptotic expansion the expression involves only the standard dynamic hyperpolarizabilities.

Third-order many-body perturbation theory calculations for the beryllium and magnesium isoelectronic sequences

Third-order relativistic many-body perturbation theory (MBPT) is applied to obtain energies of ions with two valence electrons in the no virtual-pair approximation (NVPA). A total of 302 third-order Goldstone diagrams are organized into 12 one-body and 23 two-body terms. Only third-order two-body terms and diagrams are presented in this paper, owing to the fact that the one-body terms are identical to the previously studied third-order terms in monovalent ions. Dominant classes of diagrams are identified. The model potential is the Dirac-Hartree-Fock potential ${V}^{N\ensuremath{-}2}$, and $B$-spline basis functions in a cavity of finite radius are employed in the numerical calculations. The Breit interaction is taken into account through the second order of perturbation theory, and the lowest-order Lamb shift is also evaluated. Sample calculations are performed for berylliumlike ions with $Z=4--7$, and for the magnesiumlike ion $\mathrm{P}\phantom{\rule{0.2em}{0ex}}IV$. The third-order excitation energies are in excellent agreement with measurement with an accuracy at 0.2% level for the cases considered. Comparisons are made with second-order MBPT results, and with other calculations. The third-order energy correction is shown to be significant, improving the previous second-order calculations by an order of magnitude.

Symmetry-adapted perturbation theory of three-body nonadditivity of intermolecular interaction energy

The many-body symmetry-adapted perturbation theory of intermolecular interactions has been extended to the interactions of three closed-shell atoms or molecules. Orbital formulas for the nonadditivity of the second- and third-order induction energies, third-order induction-dispersion energy, and third- and fourth-order dispersion energies with neglect of intramonomer correlation effects have been derived from the polarization expansion. The first-order intramonomer correlation correction to the third-order dispersion energy has also been obtained. The symmetrized Rayleigh–Schrödinger energy expansion has been used to derive orbital formulas for the first- and second-order exchange energies with neglect of intramonomer correlation effects. The first-order exchange energy was approximated by the terms proportional up to the fourth power in the intermolecular overlap integrals. The second-order exchange energy was developed through the third power of the intermolecular overlap integrals.

Third-order multireference perturbation theory The CASPT3 method

Rayleigh-Schrodinger perturbation theory is applied to compute second and third-order correlation energies using complete active space self-consistent field (CASSCF) zeroth-order wavefunctions. The first-order wavefunction is expanded in a basis of internally contracted configurations. The zeroth-order Hamiltonian is a sum of one-electron effective Fock operators, which are invariant to unitary transformations among the active orbitals. Comparisons with FCI data are made for the singlet-triplet splitting of CH2 and the barrier height and exothermicity of the F + H2 reaction. Potential energy functions and spectroscopic data are computed for C2, CN, CO, CF,N2, NO, O2, and F2 using large basis sets. It is demonstrated that the third-order results (CASPT3) are significantly more accurate than the second-order (CASPT2) ones. The equilibrium distances and harmonic frequencies obtained with CASPT3 are found to be as accurate as MRCI and RCCSD(T) values, while dissociation energies are generally somewhat too sma...

Spectral Theory of Physical and Chemical Binding,

A spectral method which provides unified quantum mechanical descriptions of both physical and chemical binding phenomena is reported for constructing the adiabatic electronic potential energy surfaces of aggregates of atoms or other interacting fragments. The formal development, based on use of a direct product of complete sets of atomic spectral eigenstates and the pairwise-additive nature of the total Hamiltonian matrix in this basis, is seen to be exact when properly implemented and to provide a separation theorem for N-body interaction energies in terms of response matrices which can be calculated once and for all for atoms and other fragments of interest. Its perturbation theory expansion provides a generalization of familiar (Casimir−Polder) second-order pairwise-additive and (Axilrod−Teller) third-order nonadditive interaction energies, expressions which are recovered explicitly in the long-range-dipole expansion limit. A program of ab initio computational implementation of the formal development i...

Pinnacle: An approach toward object oriented quantum chemistry

This article presents an object orientated approach to the calculation of quantum chemical quantities, specifically Moller-Plesset second- and third-order energies. Blocking and index symmetry is used to take advantage of the sparsity of the data structures. The two main classes, Arrays and BlockArrays, are presented along with a sample code segment. While this code has been specifically used to calculate quantum chemical quantities, the code has been made general enough to be of use in other computational settings. © 1995 John Wiley & Sons, Inc.

Quantum field theoretical methods in chemically bonded systems. V. Potential energy curves for N2(X1?g+) ? 2N (4S)

Many-body perturbation theory is applied to the nitrogen triple bond for bond distances ranging from the atomic regime to about 0.6a0 shorter than equilibrium. A full-optimized reaction space model is used to compute orbital spaces with an even-tempered gaussian-type basis set and also with a nominal Bagus–Gilbert Slater-type basis set. Conservation of orbital angular momentum in the atomic regime leads to perturbative theory for Hartree–Fock plus proper dissociation. Angular momentum conservation can also be enforced with a scaled Slater–Condon parameter. Third-order dissociation energies and spectroscopic constants approach limits of the chosen basis sets. © 1993 John Wiley & Sons, Inc.

Correlation energy of the free-electron pair in the second-order perturbation theory

A standard Rayleigh–Schrödinger perturbation theory is applied to a free-electron pair enclosed in a spherical potential box having infinite walls and to an itinerant free-electron pair at its lowest energy. The second-order perturbation energy for the standing-like wave functions is found to be a large, but finite negative number independent of the box dimensions. A similar energy calculated for the itinerant free-electron pair at its lowest energy level has a much smaller absolute value; also the third-order perturbation energy for the same itinerant pair is found to have a convergent value. The convergent results obtained for the second-order and third-order perturbation energies of the free-electron pair revise the view that these energies are the divergent quantities.

Relativistic many-body perturbation-theory calculations on Kr: accurate representation of second- and third-order energies with contracted Gaussian basis set

Relativistic many-body perturbation-theory calculations on Kr have been performed with basis sets of contracted and uncontracted Gaussian-type functions. Second- and third-order Coulomb correlation corrections were computed by systematically enlarging the virtual space. The contracted Gaussian basis sets used in the many-body study are designed to retain flexibility for correlated calculations both in the core and the valence region. They reproduce second- and third-order energy corrections computed with uncontracted Gaussian basis sets to an accuracy of over 99.9%. This type of contraction has the advantage that it reduces integral storage requirements and the time needed for correlated calculations, while retaining high accuracy. Non-additive contributions, due to the interference between relativity and the Coulomb correlation in the second-order correlation energy correction in Kr, is ≈ 0.013 au, an order of magnitude larger than in Ar.

'Partitions' in the perturbation theory of n-electron systems. III. The third-order energy

For pt.II, see ibid., vol.19, p.1863 (1986). In the perturbation theory of the general n-electron system, H(n)= Sigma nf(i)+ lambda Sigma ng(i,j), the subsystem method provides a finite decomposition (termed 'partition') in each order of the perturbation eigenfunctions and energies. The author shows that the third-order energy E(3)(n) is a weighted sum of third-order energies E(3)(n1) belonging to electronic states (most of them highly excited) of the n1=four-, three- and two-particle subsystems. The proof uses a parentage expansion of the second-order n-electron eigenfunction. The energy partitions can be regarded as the rigorous derivation of the important 'atomic energy relations' intuitively derived by Bacher and Goudsmit (1934).

Ab initio g-Hartree calculations on the atoms Be and Ne that employ fully numerical and basis-set techniques.

We report fully numerical calculations of correlation energies in second-order perturbation theory based on Hartree-Fock and g-Hartree mean fields for Be and Ne atoms and show that basis-set techniques give essentially the same results. An extension to higher-order perturbation theory is relatively straightforward in the basis-set approach. We report third-order calculations that yield in the case of Be significantly improved values for the g-Hartree correlation energy which now agrees within 4% with the total correlation energy known from complete configuration-interaction estimates, while third-order Hartree-Fock correlation energies calculated in the same basis set still disagree by 10% due to difficulties in describing the near degeneracy of the 2s and 2p orbitals. In the case of the Ne atom, we find that for the g-Hartree mean field still higher orders of perturbation theory are required to obtain agreement with experiment.

Structure of the correlation energy in 3d1 systems

Third-order pair energies and the total third-order energy are obtained for the first time for an atomic system containing 3d electrons. The variational–perturbation method is applied within the framework of the Rayleigh–Schrödinger perturbation theory with H0=HHF. The N-electron first-order wave function is defined in terms of orbital configurations of one-electron functions and symmetry-adapted pair functions in the form of partial-wave (PW) expansions. Extensive basis sets including all PW up to l,l′≤9 are used. For Zn2+ the total third-order energy amounts to about 9% of the second-order energy obtained in the same basis set, which is considerably more than for closed-shell ten-electron systems. Values of the energy functional for the function ψ0+λψ1 are also reported. The latter represent the most accurate variational results obtained so far for atomic systems containing 3d electrons.

Diagrammatic many-body perturbation expansion for atoms and molecules: II. Second-order and third-order ladder energies

Program obtainable from: CPC Program Library, Queen’s Oppenheimer approximation. University of Belfast, N. Ireland (see application form in Method of solution this issue) The diagrammatic many-body perturbation expansion is employed through third-order in the energy and first-order Computer: IBM 360/91; Installation: The Johns Hopkins in the wavefunction, including all many-body effects that University Applied Physics Laboratory arise. The calculations areperformed within the algebraic approximation [1] in whicheigenfunctionsare parameterOperating system: ASP izedby expansion in a finite set of basis functions. Computer algorithms are presented for the evaluation of second-order Programming language: FORTRAN IV energies; overlap integrals over the first-order perturbative wavefunction; and third-order particle--particle andhole—hole High speed storage required: 200 kilobytes ladder diagrams.