Zeroth-order Wave Function Research Articles

Energy versus amplitude corrected coupled-cluster approaches. II. Breaking the triple bond

We examine the effectiveness of various energy corrections to the standard CCSD and to the reduced multireference (RMR) CCSD methods. These corrections are based on the asymmetric energy formula, but instead of projecting onto the reference configuration, as in the standard CCSD method, we employ for this purpose either the MR CISD wave function that is based on a suitable model space of the kind used in RMR CCSD, or simply the zero-order wave function in that model space. Both full complete-active-space and severely-truncated model spaces are employed. The method is applied to the prototypical case of the triple-bond dissociation, namely, to the exactly solvable double-zeta model of the N2 molecule. It is shown that in this way we can eliminate the breakdown of the standard CCSD method in the region of highly stretched geometries and obtain reliable potential energy curves. The comparison with the recently proposed renormalized CCSD(T) and variational CCD methods is also briefly addressed.

Different viewpoints of electric dipole–electric dipole interaction in the calculation of the energy transfer rate of rare earth ions in insulators

The physical mechanism of the electric dipole–electric dipole (ED–ED) interaction may be looked at from several viewpoints in the calculation of the ED–ED energy transfer rate of rare earth ions in insulators. In this article, we demonstrate the equivalence of the results calculated from taking the ED–ED interaction as an energy transfer operation, or as a correction operator to the zero-order wavefunction of the donor–acceptor system.

Theoretical Calculations of Strain-Induced Splittings and/or Shifts of and Levels for

Under various uniaxial stresses, both strains with low symmetry and isotropic strains of crystals take place. The former gives the strain-induced low-symmetry crystal fields and accordingly splittings of levels; the latter gives the isotropic parts of strain-induced crystal fields and accordingly shifts of levels. By using the wavefunctions obtained from the diagonalization of the complete energy matrix in a regular octahedral field, the relevant matrix elements and accordingly strain-induced splittings and/or shifts of and for have been calculated. Their physical origins have been thoroughly analyzed and revealed. It is the admixtures of basic wavefunctions resulted from the spin-orbit interaction and/or Coulomb interaction and/or Kramers degeneracy that make strain-induced splittings of levels nonzero. In contrast with this, strain-induced shifts come mainly from the zero-order approximate wavefunctions. It is found that there are nonvanishing matrix elements of operators and between wavefunctions with positive and those with negative , which have important effects on strain-induced splittings of levels. The shifts of under both hydrostatic pressure and uniaxial pressure have been uniformly calculated. The important results of and have been evaluated.

Introduction of n-electron valence states for multireference perturbation theory

The present work presents three second-order perturbative developments from a complete active space (CAS) zero-order wave function, which are strictly additive with respect to molecular dissociation and intruder state free. They differ by the degree of contraction of the outer-space perturbers. Two types of zero-order Hamiltonians are proposed, both are bielectronic, incorporating the interactions between electrons in the active orbitals, therefore introducing a rational balance between the zero-order wave function and the outer-space. The use of Dyall’s Hamiltonian, which puts the active electrons in a fixed core field, and of a partially contracted formalism seems a promising compromise. The formalism is generalizable to multireference spaces which are parts of a CAS. A few test applications of the simplest variant developed in this paper illustrate its potentialities.

Performance of CCSDT for first row AB/AB− diatomics: Dissociation energies and electron affinities

CCSDT/aug-cc-pVXZ (X=D,T,Q) calculations were performed on the C2, C2−, CN, CN−, O2, and O2− first row diatomic molecules. The inclusion of diffuse functions improves the dissociation energies of the anionic systems by 2.0–3.4 kcal/mol, which is relevant bearing in mind the goal of achieving chemical accuracy. The contribution of the diffuse functions in the case of neutral O2 (0.6 kcal/mol) is by no means negligible in this context. A serious discrepancy between the theoretical prediction and the experimental values available for the dissociation energy of C2− was found. Since the theoretical deficiences commonly ascribed to the CCSDT method (single-reference and spin contamination when using UHF zeroth-order wave functions) cannot be invoked in this case, further experimental work is required to throw some light on the origin of such a discrepancy. The performance of CCSDT for adiabatic electron affinities is excellent in the case of O2/O2− and CN/CN−. For C2/C2−, the observed discrepancies can be explained in terms of the well-known multiconfigurational nature of the ground state of the C2 molecule.

Ground and valence excited states of C2N and CN2 transients: Ab initio geometries, electronic structures, and molecular properties

Geometric and vibrational characterization of CCN(X̃ 2Π,ã 4Σ−,Ã 2Δ,B̃ 2Σ−,C̃ 2Σ+), CNC(X̃ 2Πg,Ã 2Δu,B̃ 2Σu−), CNN(X̃ 3Σ−,ã 1Δ,b̃ 1Σ+,Ã 3Π,1 1Π) and NCN(X̃ 3Σg−,ã 1Δg,b̃ 1Σg+,Ã 3Πu) systems have been done using full-valence complete active space SCF (CASSCF) method. The Renner–Teller interaction parameter, ε, is calculated for Π electronic states with CASSCF potentials. Excitation energies with zero-point corrections, T0, electric field gradient (efg), and dipole moment, μ, are calculated using CASSCF, complete active space second order perturbation theory (CASPT2) and multireference singles and doubles configuration interaction (MRD-CI) levels of theory. The fact that CASSCF values of the principal components VXX, VYY, and VZZ of the efg tensor listed through two quantities eq1(=VZZ) and eq2(=VXX−VYY) are not very different from their CASPT2 counterparts, suggests that second-order perturbation involving all singles and doubles over the one-dimensional space spanned by the CASSCF wave function are not important for the efg and μ. However, the important contributions come from the higher excitations (triple, quadruples, etc.), which are included in MRD-CI wave function, by taking multireference zeroth-order wave function. The use of iterative natural orbital seems to be necessary to obtain stable values of the efg and μ in the MRD-CI method.

On a mixed Møller–Plesset Epstein–Nesbet partition of the Hamiltonian to be used in multireference perturbation configuration interaction

Within the frame of second-order multi-reference perturbation theory (MR-PT), drawing on previous work of Dyall, a new partition of the Hamiltonian, which can loosely be considered as intermediate between Møller–Plesset (MP) and Epstein–Nesbet (EN), is presented. The method, which is well suited for CAS-SCF zero-order wavefunctions, is expected to correct partially the defects of MP and EN while retaining their good properties. The new approach has been successfully applied to some test cases involving the homonuclear diatomics F 2, Cl 2 and N 2 where the faults of the EN treatment appear to be in large measure eliminated. Application to the Cr 2 diatomic shows that the dramatic intruder state problem suffered by the MP treatment is completely solved, affording a qualitatively good description of the potential energy curve for the ground state.

A multiconfigurational self-consistent field study of the thermal decomposition of methyl azide

Thermal decomposition of methyl azide has been studied computationally by using the complete active space self-consistent field (CASSCF) method and Møller–Plesset theory using the CASSCF wave function as the zeroth-order wave function (CAS/MP2). The calculations have been performed in conjunction with the 6-31G* basis set. The reaction is predicted to occur in two steps via nitrene intermediate: (1) CH3N3→CH3N+N2; (2a) CH3N→H2+HCN, (2b) CH3N→H2CNH. The rate-limiting step is the N2 extrusion (1), being a competitive mechanism between a spin-forbidden path and a spin-allowed one. The calculated energy barrier height for both processes is found to be isoenergetic, ΔE=41 kcal/mol, where ΔE represents the difference between the energy at the minimum on the singlet state surface of methyl azide and the energy at the minimum energy crossing structure (ISC1) or the singlet transition state (TS1) for the spin-forbidden path and the spin-allowed one, respectively. The nitrene intermediate formed in step (1) can undergo two parallel reactions: (2a) decomposition in H2 and HCN, and (2b) isomerization to methyleneimine. H2 extrusion from the imine generated in step (2b) has been studied as well. Two high energy transition states have been found for 1,1-H2 and 1,2-H2 elimination from methyleneimine, respectively.

Approximation of an infinite periodic system by a self-consistent embedding of a finite cluster: The dressed-cluster method

A numerical method to treat periodic spin or electron lattices is presented. It consists in considering a finite cluster embedded in a periodic strongly localized zero-order wave function. The full configuration interaction (CI) internal to the finite cluster is performed and the corresponding wave function is exploited to treat the effect of the elementary excitations occurring outside the cluster through a self-consistent dressing of the CI matrix. The method is closely related to the coupled cluster method, but of much easier use. Its efficiency is illustrated on a series of calculations on one-dimensional systems, one of them exhibiting a phase transition.

Investigation of an asymmetric triple-excitation correction for coupled-cluster energies

A correction for the effects of connected triple excitations to the coupled-cluster singles and doubles energy is studied. The approach relies on the fact that the ground-state coupled-cluster energy may be viewed as an eigenvalue of an effective (similarity transformed) Hamiltonian with associated left and right eigenvectors. Taking these as zeroth-order wave functions and using a conventional partitioning of the bare electronic Hamiltonian, the lowest order triple-excitation correction to the correlation energy is found to have an asymmetric form that involves cluster amplitudes as well as components of the left eigenvector. The popular (T) correction may be viewed as an approximation to the present approach, though the latter is approximately a factor of 2 more expensive to compute. The method is applied to a number of difficult cases, including the harmonic vibrational frequencies of ozone and the equilibrium bond length of N2. In addition, the theory of analytic gradients for the method is outlined and some aspects regarding its implementation are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 601–611, 1998

Scaling behavior of dynamic correlation effects

The authors present an analytical estimate of the intrinsic errors of second-order perturbation theory in the complete basis set limit in the framework a scaling theory, which employs the maximal sharpness of the orbitals in a given basis set, quantified by the average momentum, as its fundamental variable. The authors find that the intrinsic errors of second-order perturbation theory fall with the third inverse power of the maximal momentum representable in the basis set, while the basis set truncation errors fall with the second inverse power of the momentum. These analytical arguments are verified in a numerical investigation employing distributed basis sets of up to 600 orbitals for the helium atom, water, and ethene. Extending the analysis to generic multireference perturbation theory (MRPT) the authors find that the leading contribution for the relaxation of the orthogonal complement of the zero-order wave function in the primary space scales with the same power as the leading overall perturbative correction, which yields an analytical argument for the perturbation inclusion of such feedback effects in the formulation of efficient versions of MRPT.

Scaling behavior of dynamic correlation effects

We present an analytical estimate of the intrinsic errors of second-order perturbation theory in the complete basis set limit in the framework a scaling theory, which employs the maximal “sharpness” of the orbitals in a given basis set, quantified by the average momentum, as its fundamental variable. We find that the intrinsic errors of second-order perturbation theory fall with the third inverse power of the maximal momentum representable in the basis set, while the basis set truncation errors fall with the second inverse power of the momentum. These analytical arguments are verified in a numerical investigation employing distributed basis sets of up to 600 orbitals for the helium atom, water, and ethene. Extending our analysis to generic multireference perturbation theory (MRPT) we find that the leading contribution for the relaxation of the orthogonal complement of the zero-order wave function in the primary space scales with the same power as the leading overall perturbative correction, which yields an analytical argument for the perturbation inclusion of such feedback effects in the formulation of efficient versions of MRPT. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 147–157, 1998

Why CCSD(T) works: a different perspective

The CCSD(T) method was originally motivated as an attempt to treat the effects of triply excited determinants upon both single and double excitation operators on an equal footing. Hence, conventional analyses based on perturbation theory cannot satisfactorily explain why the particular fifth-order term included in CCSD(T) should be chosen over a number of other possibilities. This work demonstrates that the terms appearing in CCSD(T) can be justified if one takes the biorthogonal representation of the CCSD state as the zeroth-order wavefunction. This perspective provides some additional insight as to why the method works so well in practice.

An ab Initio QCISD Study of the Potential Energy Surface for the Reaction HNO + NO → N2O + OH

Results are presented from ab initio quantum chemical calculations on the ground-state potential energy surface for the open shell doublet reaction: HNO + NO → N2O + OH (reaction I). This reaction is thought to be important in the sustained combustion of nitrogen-containing fuels. Geometry optimizations have been carried out at the QCISD/6-31G(d,p) level using unrestricted Hartree−Fock zeroth-order wave functions in search of local minima and transition states, followed by energy refinements for all critical points at the QCISD(T)/6-311+G(2df,2p) level. All saddle points were subjected to intrinsic reaction coordinate calculations to determine the minima to which they connect. We report two completed reaction pathways for reaction I. Some of these results will be compared with earlier work by others on the same reaction where geometry optimizations were carried out primarily at the UMP2/6-311G(d,p) level. The comparison brings to light some interesting differences between QCISD and UMP2 theories in the prediction of molecular structures and the connectivity of critical points (and therefore, connected reaction paths) for the doublet system in reaction I.

The X˜ 1 A 1 , a˜ 3 B 1 , a˜ 1 B˜ 1 , and B˜ 1 A 1 electronic states of SiH 2

Four electronically low-lying states of silylene (SiH2) have been studied systematically using high level ab initio electronic structure theory. Self-consistent field (SCF), two-configuration (TC) SCF, complete active space (CAS) SCF, configuration interaction with single and double excitations (CISD), and CASSCF second-order (SO) CI levels of theory were employed with eight distinct basis sets. The zeroth-order wave functions of the ground (X˜ 1A1 or 1 1A1) and B˜ 1A1 (or 2 1A1) excited states are appropriately described by the first and second eigenvectors of the TCSCF secular equations. The TCSCF-CISD, CASSCF, and CASSCF-SOCI wave functions for the B˜ 1A1 (or 2 1A1) state were obtained by following the second root of the CISD, CASSCF, and SOCI Hamiltonian matrices. At the highest level of theory, the CASSCF-SOCI method with the triple zeta plus triple polarization augmented with two sets of higher angular momentum functions and two sets of diffuse functions basis set [TZ3P(2f,2d)+2diff], the energy separation (T0) between the ground (X˜ 1A1) and first excited (a˜ 3B1) states is determined to be 20.5 kcal/mol (0.890eV,7180cm−1), which is in excellent agreement with the experimental T0 value of 21.0 kcal/mol (0.910eV,7340cm−1). With the same method the T0 value for the a˜ 1B1−X˜ 1A1 separation is predicted to be 45.1 kcal/mol (1.957 eV,15780 cm−1), which is also in fine agreement with the experimental value of 44.4 kcal/mol (1.925 eV,15530 cm−1). The T0 value for the B˜ 1A1−X˜ 1A1 separation is determined to be 79.6 kcal/mol (3.452 eV,27 840 cm−1). After comparison of theoretical and experimental T0 values for the a˜ 3B1 and a˜ 1B1 states and previous studies, error bars for the B˜ 1A1 state are estimated to be ±1.5 kcal/mol (±525 cm−1). The predicted geometry of the B˜ 1A1 state is re(SiH)=1.458 and θe=162.3∘. The physical properties including harmonic vibrational frequencies of the B˜ 1A1 state are newly determined.

On the role of single excitations in quasi-degenerate perturbation theory

In multiconfiguration perturbation theory, the generalized Brill⊙uin theorem is normally fulfilled, which imposes constraints on the matrix elements involved in the description of single excitations. This can be considered as guaranteeing coherence of the configurations and orbitals describing the zeroth order wavefunction. In quasi-degenerate perturbation theory (QDPT) single excitations are treated independently, which causes a loss of coherence. It is argued that a loss of orbital coherence leads to unreasonable QDPT correlation energies.

Calculation of electronic tunneling matrix element in proteins: Comparison of exact and approximate one-electron methods for Ru-modified azurin

In recent years several theoretical methods have been developed for evaluation of the magnitude of electronic coupling between distant donor and acceptor complexes mediated by a protein molecule. Most detailed studies have been carried out within the one-electron tight-binding (extended Hückel) approximation for electronic structure of the protein medium. In this paper different approximate and exact one-electron methods such as perturbation theory, exact diagonalization, and method of tunneling currents are reviewed and results of calculations are compared for three HisX-Ru-modified azurin molecules, where X=122, 124, and 126. These systems have been recently synthesized and studied experimentally by Gray and co-workers. The calculations show that perturbation theory results are in excellent agreement with exact calculations if the symmetry of the zeroth-order wave functions of the donor and acceptor metal ions are chosen correctly. A simple computational procedure for construction of such correct zeroth-order functions is proposed.

Ab initio calculation of the electronic coupling element in bimetallic model compounds [MLM] +, M = Be, Mg, Zn; L = O, S, CH 2, CC; electron correlation effects and dependence on the bridge nature

Ab initio calculations of the electron-transfer matrix element V ab in bimetallic model systems [MM] + and [MLM] + with M = Be, Zn and L = O, S, CH 2 and CC, are reported. Electron correlation effects are estimated through a second-order method in which only the differential space contributing to the transfer integral H ab is considered. V ab is found to be noticeably larger for bridged systems, showing that the through bond mechanism is pre-eminent in these systems. In most of the calculations, electron correlation contributions are found to increase V ab , the corrections in some cases being as large as 35%. Analysis of the second-order contributions in these cases shows that intruder states involving ligand or metal ligand excitations strongly mix with the zeroth-order model wave functions.

Perturbation Theory for Quantum Mechanics in its Hessenberg-Matrix Representation

For years, the partial integrability in quantum mechanics [e.g., the exceptional terminating Lanczos solutions or the so called quasi-exactly solvable "next-to-elementary" systems] represented a challenge in perturbation theory. The main difficulty lied in an incompleteness of the available zero-order wavefunctions and in the related impossibility of an easy construction of the necessary zero-order propagators. We describe a solution of this problem, based on an unusual choice of model space. A few examples illustrate the underlying technicalities: Paying detailed attention to the Hessenberg-matrix Hamiltonians, our formalism compensates the incompleteness of integrability by a slightly less elementary (viz., non-diagonal but still triangular) matrix structure of its propagators.

Fourier transform approach to potential harmonics

Because of the high degeneracy of hyperspherical harmonics, a method is needed for selecting the most important ones for inclusion in hyperangular basis sets. Such a method was developed by M. Fabre de la Ripelle, who showed that the most important harmonics are λ-projections of the product of the potential and a zeroth-order wave function; and he gave these the name, “potential harmonics.” In the present study we develop Fourier-transform-based methods for generating potential harmonics and for evaluating matrix elements between them. These methods are illustrated by a small calculation on three-body Coulomb systems with a variety of mass ratios. © 1997 John Wiley & Sons, Inc.