Multideterminant Wave Functions Research Articles

EDF: Computing electron number probability distribution functions in real space from molecular wave functions

Given an N-electron molecule and an exhaustive partition of the real space ( R 3 ) into m arbitrary regions Ω 1 , Ω 2 , … , Ω m ( ⋃ i = 1 m Ω i = R 3 ), the edf program computes all the probabilities P ( n 1 , n 2 , … , n m ) of having exactly n 1 electrons in Ω 1 , n 2 electrons in Ω 2 , … , and n m electrons ( n 1 + n 2 + ⋯ + n m = N ) in Ω m . Each Ω i may correspond to a single basin (atomic domain) or several such basins (functional group). In the later case, each atomic domain must belong to a single Ω i . The program can manage both single- and multi-determinant wave functions which are read in from an aimpac-like wave function description ( .wfn) file (T.A. Keith et al., The AIMPAC95 programs, http://www.chemistry.mcmaster.ca/aimpac, 1995). For multi-determinantal wave functions a generalization of the original .wfn file has been introduced. The new format is completely backwards compatible, adding to the previous structure a description of the configuration interaction (CI) coefficients and the determinants of correlated wave functions. Besides the .wfn file, edf only needs the overlap integrals over all the atomic domains between the molecular orbitals (MO). After the P ( n 1 , n 2 , … , n m ) probabilities are computed, edf obtains from them several magnitudes relevant to chemical bonding theory, such as average electronic populations and localization/delocalization indices. Regarding spin, edf may be used in two ways: with or without a splitting of the P ( n 1 , n 2 , … , n m ) probabilities into α and β spin components. Program summary Program title: edf Catalogue identifier: AEAJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5387 No. of bytes in distributed program, including test data, etc.: 52 381 Distribution format: tar.gz Programming language: Fortran 77 Computer: 2.80 GHz Intel Pentium IV CPU Operating system: GNU/Linux RAM: 55 992 KB Word size: 32 bits Classification: 2.7 External routines: Netlib Nature of problem: Let us have an N-electron molecule and define an exhaustive partition of the physical space into m three-dimensional regions. The edf program computes the probabilities P ( n 1 , n 2 , … , n m ) ≡ P ( { n p } ) of all possible allocations of n 1 electrons to Ω 1 , n 2 electrons to Ω 2 , … , and n m electrons to Ω m , { n p } being integers. Solution method: Let us assume that the N-electron molecular wave function, Ψ ( 1 , N ) , is a linear combination of M Slater determinants, Ψ ( 1 , N ) = ∑ r M C r ψ r ( 1 , N ) . Calling S Ω k r s the overlap matrix over the 3D region Ω k between the (real) molecular spin-orbitals (MSO) in ψ r ( χ 1 r , … χ N r ) and the MSOs in ψ s , ( χ 1 s , … , χ N s ) , edf finds all the P ( { n p } ) 's by solving the linear system (1) ∑ { n p } { ∏ k m t k n k } P ( { n p } ) = ∑ r , s M C r C s det [ ∑ k m t k S Ω k r s ] , where t m = 1 and t 1 , … , t m − 1 are arbitrary real numbers. Restrictions: The number of { n p } sets grows very fast with m and N, so that the dimension of the linear system (1) soon becomes very large. Moreover, the computer time required to obtain the determinants in the second member of Eq. (1) scales quadratically with M. These two facts limit the applicability of the method to relatively small molecules. Unusual features: Most of the real variables are of precision real*16. Running time: 0.030, 2.010, and 0.620 seconds for Test examples 1, 2, and 3, respectively.

Energies of the first row atoms from quantum Monte Carlo

All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.

Higher-order equation-of-motion coupled-cluster methods for electron attachment

High-order equation-of-motion coupled-cluster methods for electron attachment (EA-EOM-CC) have been implemented with the aid of the symbolic algebra program TCE into parallel computer programs. Two types of size-extensive truncation have been applied to the electron-attachment and cluster excitation operators: (1) the electron-attachment operator truncated after the 2p-1h, 3p-2h, or 4p-3h level in combination with the cluster excitation operator after doubles, triples, or quadruples, respectively, defining EA-EOM-CCSD, EA-EOM-CCSDT, or EA-EOM-CCSDTQ; (2) the combination of up to the 3p-2h electron-attachment operator and up to the double cluster excitation operator [EA-EOM-CCSD(3p-2h)] or up to 4p-3h and triples [EA-EOM-CCSDT(4p-3h)]. These methods, capable of handling electron attachment to open-shell molecules, have been applied to the electron affinities of NH and C2, the excitation energies of CH, and the spectroscopic constants of all these molecules with the errors due to basis sets of finite sizes removed by extrapolation. The differences in the electron affinities or excitation energies between EA-EOM-CCSD and experiment are frequently in excess of 2 eV for these molecules, which have severe multideterminant wave functions. Including higher-order operators, the EA-EOM-CC methods predict these quantities accurate to within 0.01 eV of experimental values. In particular, the 3p-2h electron-attachment and triple cluster excitation operators are significant for achieving this accuracy.

Electron number probability distributions for correlated wave functions

Efficient formulas for computing the probability of finding exactly an integer number of electrons in an arbitrarily chosen volume are only known for single-determinant wave functions [E. Cances et al., Theor. Chem. Acc. 111, 373 (2004)]. In this article, an algebraic method is presented that extends these formulas to the case of multideterminant wave functions and any number of disjoint volumes. The derived expressions are applied to compute the probabilities within the atomic domains derived from the space partitioning based on the quantum theory of atoms in molecules. Results for a series of test molecules are presented, paying particular attention to the effects of electron correlation and of some numerical approximations on the computed probabilities.

Excited states of boron isoelectronic series from explicitly correlated wave functions

The ground state and some low-lying excited states arising from the 1s2 2s2p2 configuration of the boron isoelectronic series are studied starting from explicitly correlated multideterminant wave functions. One- and two-body densities in position space have been calculated and different expectation values such as <delta(r)>, <rn>, <delta(r12)>, <r12 n>, <delta(R)>, and <R(n)>, where r, r12, and R stand for the electron-nucleus, interelectronic, and two electron center of mass coordinates, respectively, have been obtained. The energetic ordering of the excited states and the fulfillment of the Hund's rules is analyzed systematically along the isoelectronic series in terms of the electron-electron and electron-nucleus potential energies. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding noncorrelated ones. All the calculations have been done by using the variational Monte Carlo method.

Using on-top pair density for construction of correlation functionals for multideterminant wave functions

The value of the two-particle density function at coalescence is frequently used as an additional variable for formulating approximate exchange-correlation or correlation functionals. Here, its applications as one of the key variables for the construction of new DFT (preferably multi-determinant) functionals is investigated. The basic formalism is presented and it is shown that this replacement avoids some difficulty to construct a Fock matrix in a ROKS (restricted open-shell Kohn–Sham) method and also to reduce the ‘double counting’ of correlation energy in CASDFT (complete active space density functional theory) calculations. Calculations of excitation energies for transition metals and dissociation curves for diatomic molecules are presented as an example.

Density functional theory with alternative spin densities: application to magnetic systems with localized spins.

A new method to improve the excess spin density obtained from unrestricted Hartree-Fock wave functions in terms of natural orbitals is proposed. Using this modified excess spin density to evaluate the correlation energy by means of density functionals leads to large improvements in the computed magnetic coupling constants of several materials without need to modify the exchange contribution. This is important because it reconciles the density functional theory description with the one provided by multi-determinant wave functions. Using the present approach, the leading contribution to the magnetic coupling constant arises from electron correlation effects. The performance of the new method is illustrated on various materials including high-critical-temperature superconductors parent compounds.

Local Spin III: Wave Function Analysis along a Reaction Coordinate, H Atom Abstraction, and Addition Processes of Benzyne

The recently derived local spin operator 〈SA2〉 directly determines the spin state of an atom or molecular fragment A, whereas the 〈SA·SB〉 operator represents the Heisenberg Hamiltonian spin coupling between A and B. Although one typically associates such spin properties with open-shell molecules, in the single-determinant molecular orbital approximation, these operators may be related to chemically relevant quantities such as bond order regardless of whether the system has unpaired electrons. Here we demonstrate the usefulness of these operators as molecular properties that can be applied to wave functions for which many properties are not defined, namely, multideterminant wave functions. Analysis of the wave functions of o-, m-, and p-benzyne indicates that these spin operators are able to detect subtle differences in electronic structure within a series of isomers with unpaired electrons. Hydrogen atom addition and abstraction processes are then used to illustrate that the operators are able to track ch...

Variational Monte Carlo calculations for some cations and anions of the first‐row atoms using explicitly correlated wave functions

AbstractThe variational Monte Carlo method is applied to calculate ground‐state energies of some cations and anions of the first‐row atoms. Accurate values providing between 80 and 90% of the correlation energy are obtained. Explicitly correlated wave functions including up to 42 variational parameters are used. The nondynamic correlation due to the 2s − 2p near degeneracy effect is included by using a multideterminant wave function. The variational free parameters have been fixed by minimizing the energy that has shown to be a more convenient functional than the variance of the local energy, which is the most commonly employed method in variational Monte Carlo calculations. The energies obtained improve previous works using similar wave functions. © 2002 Wiley Periodicals, Inc.; DOI 10.1002/qua.10125

Momentum space densities for the beryllium isoelectronic series

One- and two-body densities in momentum space have been calculated for the atomic beryllium isoelectronic series starting from explicitly correlated multideterminant wave functions. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding Hartree–Fock ones. Some expectation values such as 〈δ(p⃗)〉, 〈pn〉, 〈δ(p⃗12)〉, 〈p12n〉, 〈δ(P⃗)〉, and 〈Pn〉, where p⃗, p⃗12, and P⃗ stand for the electron–nucleus, interelectronic, and two-electron center-of-mass momentum coordinates, respectively, and the angular correlation coefficient have been obtained. All the calculations have been carried out by using the Monte Carlo algorithm.

Maclaurin expansion of electron-pair densities in momentum space

Maclaurin expansions of the spherically averaged electron-pair intracule (relative motion) h̄( ν) and extracule (center-of-mass motion) d̄( P) densities in momentum space are studied for multi-determinant wave functions. The densities h̄( ν) and d̄( P) are expanded by even powers of ν and P, respectively. Accurate Hartree–Fock values of the derivatives h̄ (2 m) (0) and d̄ (2 m) (0) ( m=0–3) are determined for the 53 atoms from He to Xe in their ground states. The signs of the second derivatives are in agreement with the modality of h̄( ν) and d̄( P) reported in the literature, and the array of the signs of the successive derivatives is found to be intimately related with the valence electron configuration.

Optimal orbitals from energy fluctuations in correlated wave functions

A quantum Monte Carlo method of determining Jastrow–Slater and correlated multideterminant wave functions for which the energy is stationary with respect to variations in the single-particle orbitals is presented. A potential is determined by a least-squares fitting of fluctuations in the energy with a linear combination of one-body operators. This potential is used in a self-consistent scheme for the orbitals whose solution ensures that the energy of the correlated wave function is stationary with respect to variations in the orbitals. The method is feasible for atoms, molecules, and solids and is demonstrated for the beryllium, carbon, and neon atoms and for the solid diamond.

One- and two-body densities for the beryllium isoelectronic series

One- and two-body densities in position space have been calculated for the atomic beryllium isoelectronic series starting from explicitly correlated multideterminant wave functions. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding Hartree–Fock ones. Some expectation values such as 〈δ(r)〉, 〈rn〉, 〈δ(r12)〉, 〈r12n〉, 〈δ(R)〉, and 〈Rn〉, where r, r12, and R stand for the electron–nucleus, interelectronic, and two electron center of mass coordinates, respectively, have been obtained. All the calculations have been carried out by using the Monte Carlo algorithm.

A variational Monte Carlo study of the 2s-2p near degeneracy in beryllium, boron, and carbon atoms

We apply the variational Monte Carlo method to study the beryllium, boron, and carbon atoms. An explicitly correlated wave function is used in order to include the dynamic correlation among the electrons. The nondynamic correlation due to the 2s-2p near degeneracy effect present in these atoms is taken into account by using a multideterminant wave function.

Tunneling currents in long-distance electron transfer reactions. III. Many-electron formulation

Many-electron formulation of the method of interatomic tunneling currents introduced in our earlier work [J. Chem. Phys. 104, 8424 (1996); 105, 10819 (1996)] for the description of long-range electron tunneling in large molecules such as proteins or DNA is proposed. The tunneling currents can be used both for calculation of the tunneling matrix element and for the description of the spatial distribution of tunneling pathways at the atomic level of resolution. It is shown that the tunneling currents can be expressed as a matrix element of a certain (current) operator evaluated between two diabatic nonorthogonal one- or multideterminant wave functions of the initial and final states of the electrons in the system. These states can be found in the standard ground state energy minimization calculations. Explicit expressions for the currents in terms of the atomic basis functions and the transformation matrices to molecular orbitals of the donor and acceptor states are given. Thus, the proposed theory provides a method that allows ordinary electronic structure calculations to be utilized for studies of tunneling dynamics in many-electron systems. All electron–electron interactions are included in the expressions for currents at the Hartree–Fock level, so that electron polarization effects arising due to interaction of the tunneling electron and other electrons in the system are taken into account in such a description.

Correlation potentials for the He atom and the hydrogen molecule: A comparison between the correlation factor approach and DFT correlation energy functionals

Two points about correlation potentials have been dealt with in this article. The first one is related to the shape of some of the most representative correlation potentials applied to the ground state of the He atom. It is shown here that both LDA and two-body density correlation potentials compare well with that obtained through the quantum chemistry definition of correlation energy. This is an interesting result because, in previous works, it had been shown that none of the correlation potentials compared well with the Kohn–Sham one. The gradient-corrected correlation potentials exhibit a very different behavior to that of both exact potentials (quantum chemistry and Kohn–Sham ones). The other question posed here refers to how a reference to the two-body density must modify DFT functionals for the correlation energy, when a multideterminant wave function is needed. This question has been addressed by analyzing the variation of correlation potentials as the bond length of the H2 molecule increases. The results show that an external reference to the two-body density qualitatively improves DFT correlation potentials and also that only those functionals explicitly depending on two-body density can give the quantitative correct trends. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 143–156, 1998

Electronic-structure multiconfiguration calculation of a small cluster embedded in a local-density approximation host

The present paper proposes a method for electronic-structure calculations on a type of system that cannot be handled by present methods. It considers a system where a multideterminant wave function is essential for an atom or a small cluster of atoms embedded in a large system, normally a solid, which can be treated by density-functional methods such as with the local-density approximation (LDA). A suitable example is a transition-metal atom in a semiconductor or MgO host. In this method the embedding potential for the cluster is generated from a LDA calculation but applied in a multiconfiguration calculation. The method and the concept of the embedding potential are validated by application to a simple system of a cluster Li${}_{2}$Mg${}_{2}$ of four pseudoatoms.

Density functionals for the Yukawa electron‐electron interaction

AbstractShort‐range nonclassical electron‐electron interaction is described by a density functional in a scheme that allows multideterminant wave functions. The parameter that determines the coupling with the configuration‐interaction‐type calculations can be chosen in a controlled manner. Results are presented for the He and the Be series using a Yukawa‐type interaction. © 1995 John Wiley &amp; Sons, Inc.

Formally tetravalent cerium and thorium compounds: a configuration interaction study of cerocene Ce(C8H8)2 and thorocene Th(C8H8)2 using energy-adjusted quasirelativistic ab initio pseudopotentials

Abstract Large-scale state-averaged multi-configuration self-consistent field, configuration interaction, averaged coupled-pair functional and spin-orbit configuration interaction calculations have been carried out for the ground states and low-lying excited states of the bis(cyclooctatetraene)f-element sandwich complexes cerocene Ce(C 8 H 8 ) 2 and thorocene Th(C 8 H 8 ) 2 . Whereas for a single-determinant wavefunction thorocene may be pictured as a Th IV compound, i.e. a Th 4+ closed-shell ion complexed by two aromatic C 8 H 8 2− ligands, cerocene is to a first approximation a Ce III compound, i.e. a Ce 3+ ion with a 4f 1 configuration and two C 8 H 8 1.5− ligands. When cerocene is described by a multi-determinant wavefunction admixture of the Ce 4+ (C 8 H 8 2− ) 2 configuration leads to a 1 A 1g ground state of the same symmetry as in thorocene. For both compounds results for the metal-ring distance, the symmetric metal-ring stretching frequency, the photoelectron spectrum and the optical spectrum are compared to experimental data.

Ab initio studies of diamond(100) surface reconstruction.

The reconstruction of the C(100) surface is studied by a cluster model at several theoretical levels. It is found that the calculated surface-dimer bond length is very sensitive to the level of theoretical treatment, the spin state, and the degree of constraints in the geometry optimization process. A single-determinant self-consistent field (SCF) treatment gives a closed-shell singlet state, higher in energy than the triplet state, and with a dimer length of 1.401 \AA{}, 0.279 \AA{} shorter than the triplet. The dimer is found to be symmetric for both singlet and triplet states at the SCF and complete active space (2,2) levels of theory. The correct ground state is a singlet, but a multideterminant wave function is required for its description. At the configuration-interaction level, the surface-dimer bond length in the ground state is found to be 1.508 \AA{} and the energy decrease on dimer formation with respect to the ideal C(100) 1\ifmmode\times\else\texttimes\fi{}1 surface is 2.28 eV per dimer.