Orthogonal Geminals Research Articles

Density functional model of multireference systems based on geminals

A density functional model based on the variationally optimized spin-unrestricted multiconfigurational wavefunction created from strongly orthogonal singlet-type electron geminals is presented. A rescaled correlation-only PBE functional is used to account for dynamic correlation absent in the geminal wavefunction. The performance of the model is assessed from geometry optimizations on the G2/97 test set diatomics, using two different basis sets. Preliminary results presented here are encouraging and clearly warrant further studies into types of functionals that can be used in combination with the geminal reference.

Composite particles in quantum chemistry: From two‐electron bonds to cold atoms

AbstractWe review the problem of forming composite particles from elementary electrons. Simple pairs are treated first, followed by linear combinations including geminal type wave functions. The antisymmetrized product of strongly orthogonal geminals (APSG) theory can also be developed in this language, and the use of APSG as a reference state is discussed. The theory forming composite particles from more than two electrons is less elaborated, although this idea is routinely used in approximate models in the current literature. © 2012 Wiley Periodicals, Inc.

Accelerating convergence in the antisymmetric product of strongly orthogonal geminals method

AbstractAntisymmetric product of strongly orthogonal geminals (APSG) method is a wavefunction theory that can effectively treat the static electron correlation using two‐electron wavefunctions, called geminals. However, the APSG method has the problem of slow convergence in the optimization of the geminal function. In this study, we introduced the direct inversion in the iterative subspace (DIIS) method, for both closed‐ and open‐shell systems, for accelerating its convergence. Two types of error vectors, that is, unitary transformation and orbital gradient, were examined for the DIIS procedure. Numerical assessments revealed that the orbital‐gradient error vector shows better performance than the unitary‐transformation one. © 2012 Wiley Periodicals, Inc.

Separation of strong (bond-breaking) from weak (dynamical) correlation

A CC (coupled-cluster) ansatz based on a GVB (generalized valence bond) or an APSG (antisymmetrized product of strongly orthogonal geminals) reference function arises naturally if one tries to treat strong correlations exactly (to infinite order), and weak correlations by TCC (traditional coupled cluster) theory. This ansatz is proposed as an alternative to MC-CC (multi-configuration coupled cluster) theory. One uses especially that APSG and GVB are of CC type, but allow to combine separability with the variation principle. The energy and the stationarity conditions are formulated in terms of spinfree density cumulants. The replacement operators corresponding to the APSG ansatz generate a Lie algebra which is a subalgebra of that of all replacement operators.

Strongly orthogonal geminals: size-extensive and variational reference states

Properties and some applications of strongly orthogonal geminals (APSG) are reviewed emphasizing the motivations for their use and their shortcomings. An overview presents some techniques capable of improving the APSG function.

A stationary property of the APSG wave function

The method of antisymmetrized product of strongly orthogonal geminals (APSG) emerges by optimizing the one-electron functions used to construct two-electron functions (geminals), the latter being expanded (with coefficients selected variationally) in mutually exclusive subspaces of the former ones. Accordingly, E APSG is stationary with respect to the expansion coefficients and to unitary transformations of the one-electron orbitals. We show that the APSG energy is also stationary to the unitary transformation of identical geminals. For non-identical geminals, this statement holds only approximately.

An approximate wave function involving geminals of mixed spin

It has been shown by Kapuy that any alternant molecular orbital wave function is an antisymmetrized product of strongly orthogonal geminals, in which the geminals are of mixed spin. Employing the localized alternant orbitals previously introduced by the author, this fact is used to generate a wave function which is capable of interpolating between the single-parameter AMO wave function and a wave function of the Hurley, Lennard-Jones, and Pople type. The validity of the unprojected form of the least flexible wave function of this type is tested in a model calculation on the hexagonal ring of hydrogen atoms. In the test case, this unprojected wave function is found to do significantly better than the unprojected AMO wave function and almost as well as the projected AMO wave function. It is shown that in the case of an infinite system, this wave function must do better than the projected AMO wave function. Features of the more general wave functions of this type are discussed, and it is mentioned that any of these wave functions may be written as an antisymmetrized geminal powers (AGP) wave function in which the geminal is of mixed spin.

Geminal model chemistry II. Perturbative corrections.

We introduce and investigate a chemical model based on perturbative corrections to the product of singlet-type strongly orthogonal geminals wave function. Two specific points are addressed (i) Overall chemical accuracy of such a model with perturbative corrections at a leading order; (ii) Quality of strong orthogonality approximation of geminals in diverse chemical systems. We use the Epstein-Nesbet form of perturbation theory and show that its known shortcomings disappear when it is used with the reference Hamiltonian based on strongly orthogonal geminals. Application of this model to various chemical systems reveals that strongly orthogonal geminals are well suited for chemical models, with dispersion interactions between the geminals being the dominant effect missing in the reference wave functions.

Partitioning in multiconfiguration perturbation theory

The behavior of electronic wave functions within the antisymmetrized product of strongly orthogonal geminals (APSG) approximation at the point of the electron-electron coalescence is examined. Its elucidation allows one to understand the origins of the previously reported satisfactory performance of APSG wavefunctions as reference states in the connected moments expansion method (CMX). It is shown that the failure to satisfy the electron-electron cusp condition causes divergence of certain type of integrals that enter the CMX formulae, which in turn brings about significant deterioration of the quality of the resulting estimates for the ground-state energy in finite basis sets.

Behavior of the APSG electronic wavefunction near the electron-electron coalescence point

The behavior of electronic wave functions within the antisymmetrized product of strongly orthogonal geminals (APSG) approximation at the point of the electron-electron coalescence is examined. Its elucidation allows one to understand the origins of the previously reported satisfactory performance of APSG wavefunctions as reference states in the connected moments expansion method (CMX). It is shown that the failure to satisfy the electron-electron cusp condition causes divergence of certain type of integrals that enter the CMX formulae, which in turn brings about significant deterioration of the quality of the resulting estimates for the ground-state energy in finite basis sets.

Behavior of the APSG electronic wavefunction near the electron‐electron coalescence point

AbstractThe behavior of electronic wave functions within the antisymmetrized product of strongly orthogonal geminals (APSG) approximation at the point of the electron‐electron coalescence is examined. Its elucidation allows one to understand the origins of the previously reported satisfactory performance of APSG wavefunctions as reference states in the connected moments expansion method (CMX). It is shown that the failure to satisfy the electron‐electron cusp condition causes divergence of certain type of integrals that enter the CMX formulae, which in turn brings about significant deterioration of the quality of the resulting estimates for the ground‐state energy in finite basis sets.

Density matrix functional theory of four-electron systems

An approximate expression for the electron–electron repulsion energy of a closed-shell four-electron system in terms of the Coulomb and exchange integrals among natural orbitals and the respective occupation numbers is derived. It constitutes a strict upper bound to the exact density matrix functional, yields energy that is lower than that obtained within the antisymmetrized product of strongly orthogonal geminals theory, and thus is exact for two noninteracting two-electron systems. Its relevance to the general case of closed-shell N-electron systems is discussed.

Approximate one-matrix functionals for the electron–electron repulsion energy from geminal theories

A simple extension of the antisymmetrized product of strongly orthogonal geminals theory produces a “JK-only” one-matrix functional for the electron–electron repulsion energy of a closed-shell system that is exact for two-electron singlet ground states, size-extensive, and incorporates some intergeminal correlation and thus dispersion effects. The functional is defined only for one-matrices with occupation numbers that can be arranged into sets with elements that sum up to two. Its possible generalizations are discussed.

A geminal model chemistry

We modify the existing model of the antisymmetrized product of strongly orthogonal geminals to define a size consistent variational theory free from any adjustable parameters apart from the usual choice of a basis set. The new theory is only slightly more computationally expensive than Hartree–Fock, gives an exact solution for an ensemble of noninteracting singlet two electron systems, and is applicable to open- and closed-shell systems. We use it to calculate equilibrium geometries, vibrational frequencies, and dipole moments of diatomic molecules from G2/97 database. We find excellent agreement with experimental values for covalently bound molecules, and overstretched bonds of molecules formed by atoms with extreme electronegativity.

Two-body zeroth order Hamiltonians in multireference perturbation theory: The APSG reference state

A special version of multi-reference perturbation theory is investigated which differs from standard ones by using a zeroth order Hamiltonian that contains two-electron terms explicitly. The method is applicable to reference states that can be written as an antisymmetrized product of two or more electron functions. In that case the zeroth order Hamiltonian has a well defined physical meaning and the matrix elements that come about can be evaluated in an efficient manner. We implemented the theory for the antisymmetrized product of strongly orthogonal geminals wave function and, as a special case, for the generalized valence bond. Illustrative calculations on sample molecules show the reliability of the approach, as well as a significant improvement in many cases compared to MRPT versions based on one-body zeroth order Hamiltonians.

Interaction of chemical bonds. V. Perturbative corrections to geminal-type wave functions

Antisymmetrized products of strongly orthogonal geminals (APSG) and approximations thereof, provide a variational and size-consistent wave function but do not account for a sufficient amount of dynamical electron correlation. The latter effects can only be calculated by considering intergeminal interactions. In this study we present a general scheme for computing delocalization effects involving simultaneous interactions of two, three, and four geminals. We also describe a simple perturbative procedure, the APSG-MP2 method, by which APSG-type wave functions and the associated energies can be improved at a very low computational expense. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 96–104, 2000

Improving CISD calculations by geminal-type reference states

The possibility of using geminal-type wavefunctions as multi-configurational reference states for configuration interaction (CI) calculations is investigated. The reference function class considered here includes 2-electron CAS, certain types of MC-SCF and APSG (antisymmetrized product of strongly orthogonal geminals) as special cases. The method is not variational but exact for two electrons. The relation between the developed formalism and GVB-CI, internally contracted multi-reference CI (IC-MRCI) and EOM-CC techniques is discussed. We present applications at the CISD level. The results show that total energies generally fall between standard CISD and CCSD values. The proposed modification of CISD decreases the size-consistency error and eliminates it if the molecule is fragmented into two-electron pieces. Preliminary calculations show an appreciable improvement of CISD dissociation energies.

Cumulant expansion of the reduced density matrices

k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.

Nonconventional partitioning of the many-body Hamiltonian for studying correlation effects

For the treatment of electron correlation, one most often uses the Møller–Plesset (MP) partition which defines the zero-order Hamiltonian through the spectral resolution of the Fockian. We investigate how the MP partitioning can be improved while still using the Hartree–Fock (HF) reference state; and how the HF wave function can be substituted by a correlated one preserving the formal simplicity of the HF-based approach. To improve the MPn result, we introduce a fine tuning of energy denominators replacing the HF orbital energies with the ionization potentials obtained from the second-order Dyson equation. As this equation usually tends to close the gaps, a slight decrease of the denominators is expected, inducing an improvement of low-order correlation energies. We keep the simplicity of the MP partitioning and handle Dyson corrections as simple level shifts. Substituting doubly filled HF orbitals by strongly orthogonal geminals, one introduces a correlated reference state which is variational, size-consistent, and properly describes single-bond dissociation. This wave function, the antisymmetrized product of strongly orthogonal geminals (APSG), offers a good starting point for further corrections. We show that the use of an APSG reference state in the equation-of-motion technique leads to Tamm–Dankoff approach (TDA) equations which account for correlation effects in electronic excitation energies. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 571–581, 1998

Energetics and structure of Peierls-Hubbard chains in ground and excited states.

The stationary states of a half-filled Peierls-Hubbard chain are calculated. To describe electron correlations, the wave functions as antisymmetrized products of strongly orthogonal geminals are used; wave functions and bond lengths are determined self-consistently by the variational method in an N=50 ring. The neutral soliton pair ${S}^{0}$${S}^{0}$ is obtained as corresponding to a local total-energy minimum above the perfectly dimerized ground state. The charged soliton pair ${S}^{+}$${S}^{\mathrm{\ensuremath{-}}}$ refers to the minimum in the subspace of wave functions which are odd with respect to charge conjugation. The creation energies for each of the pairs are estimated as functions of Hubbard parameter U and compared with the exact excitation energies of the uniform Hubbard chain. The electron correlation is shown to make the in-gap ${S}^{0}$${S}^{0}$ state stable even for a moderate electron-lattice coupling; the soliton width turns out to be much smaller than that in the U=0 model and comparable with the lattice constant. Applicability of the results to polyacetylene is discussed.