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.     

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 .     

Near-degeneracy effects and efficiency of MBPT calculations of molecular properties. Polarizability curve of H2

The polarizability curve of H₂ is calculated by using the finite-field perturbation method. All self-consistency effects are accounted for at the HF level and many-body perturbation theory (MBPT) is used to evaluate the correlation contributions. Using a single HF determinant as a reference in MBPT calculations makes the near-degeneracy effects of essential importance on increasing the interatomic distance. Nevertheless, applying the MBPT scheme with appropriate fourth-order terms gives nearly exact values of both components of the polarizability tensor for interatomic distances up to ≈3.6 au.     

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.     

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     

Evaluation of the 3rd Order Perturbations for the Degenerate Cases

In the calculation of quantum systems in most cases, the perturbation theory is used. Lot of calculation techniques are used for both (nondegenerate and degenerate) cases. In our contribution we evaluate the projection operator technique in such case when the Hamiltonian of the given quantum system has point spectrum with finite degree of degeneracy.     

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.     

A nondiagonal quasidegenerate fourth-order perturbation theory

A canonical quasidegenerate Rayleigh-Schrödinger perturbation theory, correct through fourth order in the energy, is explored for a block-diagonal unperturbed Hamiltonian. The theory is developed completely within a Lie Algebra in Hilbert space. Explicit equations forn-particle transition elements in terms of solutions of simultaneous linear equations are presented. A two-dimensional anisotropic anharmonic oscillator is used to provide numerical results. The perturbation theory is shown to be stable under small separation of model and complement spaces. An iterative variant of the fourth order perturbation theory is introduced; the iterative variant is related to the non-iterative one in much the same way as nondegenerate coupled cluster theories are related to nondegenerate perturbation theory. The quasidegenerate coupled cluster theory appears to be stable in the presence of multiple intruder states.     

Convergence property of multireference many-body perturbation theory analyzed by the use of a norm of effective Hamiltonian

Summary It is proposed to use a norm of anth order effective Hamiltonian, for analyzing the convergence property of the multireference many-body perturbation theory (MR-MBPT). The utilization of the norm allows us to employ only (1) asingle number for all the states that we are interested in, and (2) values which decreases from thepositive side to zero as the ordern of the perturbation increases. This characteristic features are in contrast to those in the usually used scheme whereseveral numbers, namely, the eigenvalues of the target states, should be used and they mayoscillate around exact eigenvalues. The present method has been applied to MR-MBPT calculations of the (H₂)₂, CH₂, and LiH molecules based on the multireference versions of Rayleigh-Schrödinger PT, Kirtman-Certain-Hirschfelder PT, and the canonical Van Vleck PT; and following features are found: (1) the above three versions of the perturbation theories have essentially the same convergence property judged from the lowering of the norm; (2) the lower order truncation of the perturbation series gives reasonable solutions; (3) the norm decreases irrespective of the perturbation expansion being convergent or divergent for the first several orders (up to about the sixth order).     

Is fifth-order MBPT enough?

Fifth- and higher-order MBPT results are reported for a series of examples, BH, Be₂, HF and H₂O, for which higher-order perturbation theory might be important. MBPT(5) differs from MBPT(4) by as much as 4.3 mh, and by constructing the size-extensive [2,1] Padé approximant, which is possible with E⁽⁵⁾, one can get exceptional agreement with the full CI results. Variational perturbation results are also reported.     

Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond

Density functional theory (DFT) results are mistrusted at times due to the presence of an unknown exchange correlation functional, with no practical way to guarantee convergence to the right answer. The use of a known exchange correlation functional based on wave-function theory helps to alleviate such mistrust. The exchange correlation functionals can be written exactly in terms of the density-density response function using the adiabatic-connection and fluctuation-dissipation framework. The random phase approximation (RPA) is the simplest approximation for the density-density response function. Since the correlation functional obtained from RPA is equivalent to the direct ring coupled cluster doubles (ring-CCD) correlation functional, meaning only Coulomb interactions are included, one can bracket RPA between many body perturbation theory (MBPT)-2 and CCD with the latter having all ring, ladder, and exchange contributions. Using an optimized effective potential strategy, we obtain correlation potentials corresponding to MBPT-2, RPA (ring-CCD), linear-CCD, and CCD. Using the suitable choice of the unperturbed Hamiltonian, Kohn-Sham self-consistent calculations are performed. The spatial behavior of the resulting potentials, total energies, and the HOMO eigenvalues are compared with the exact values for spherical atoms. Further, we demonstrate that the self-consistent eigenvalues obtained from these consistent potentials used in ab initio dft approximate all principal ionization potentials as demanded by ionization potential theorem.     

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.     

Applicability of non-degenerate many-body perturbation theory to quasidegenerate electronic states: A model study

The nondegenerate finite-order many-body perturbation theory is applied to simple model systems in which the degree of quasidegeneracy can be continuously varied over a wide range. Three ab initio minimum basis set models involving four hydrogen atoms in various spatial arrangements are considered. The results are compared with the exact full configuration-interaction approach, double-excitation configuration-interaction and the coupled-pair many-electron theory.     

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.     

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 real-space perturbation theory for electronic correlation

We introduce a method for treating electronic correlation in which a correlation factor is optimized using Hylleraas variational perturbation theory. The factor is defined by a number of parameters which grows only linearly with system size. We test the theory on two-electron atoms using the shielded-nucleus Hamiltonian at zeroth order, obtaining −2.9035 a.u. for helium. The convergence of the method is investigated, and energies and intracule densities are compared with accurate variational results. The application of the theory to an N-electron problem with a Hartree–Fock Hamiltonian at zeroth order is 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.     

Perturbative calculation of intermolecular interactions in orthogonalized or biorthogonal basis sets

Two versions of a many-body perturbation theory for the computation of molecular interaction energies are investigated. The methods are based on the partitioning of the second-quantized form of the dimer Hamiltonian written either in the orthogonalized basis of the monomer MOs, or, alternatively, in the original non-orthogonal dimer basis set handling the overlap by the biorthogonal formalism. The zeroth-order Hamiltonian H ⁰ is the sum of effective monomer Fockians and the zeroth-order wave functions are exact eigenfunctions of H ⁰. Full antisymmetry is ensured by the second-quantized formalism. Basis set superposition error is accounted for by the counterpoise correction recipe. Results for He₂, (H₂)₂ and (H₂O)₂ indicate the reliability of the biorthogonal technique.     

On the characterization of three state conical intersections: a quasianalytic theory using a group homomorphism approach

In this work, degenerate perturbation theory through second order is used to characterize the vicinity of a three state conical intersection. This report extends our recent demonstration that it is possible to describe the branching space (in which the degeneracy is lifted linearly) and seam space (in which the degeneracy is preserved) in the vicinity of a two state conical intersection using second order perturbation theory. The general analysis developed here is based on a group homomorphism approach. Second order perturbation theory, in conjunction with high quality ab initio electronic structure data, produces an approximately diabatic Hamiltonian whose eigenenergies and eigenstates can accurately describe the three adiabatic potential energy surfaces, the interstate derivative couplings, and the branching and seam spaces in their full dimensionality. The application of this approach to the minimum energy three state conical intersection of the pyrazolyl radical demonstrates the potential of this method. A Hamiltonian comprised of the ten characteristic (linear) parameters and over 300 second order parameters is constructed to describe the branching space associated with a point of conical intersection. The second order parameters are determined using data at only 30 points. In the vicinity of the conical intersection the energy and derivative couplings are well reproduced and the singularity in the derivative coupling is analyzed.