Second-order exchange-induction energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new formulation of the second-order exchange-induction energy of symmetry-adapted perturbation theory is presented. In the proposed formalism the exchange-induction energy is expressed through one- and two-particle reduced density matrices of monomers, which are of zeroth and first order with respect to the effective electrostatic potential of another monomer. The resulting expression is further modified by using the partition of two-particle density matrices into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. The proposed formalism has been applied to the case of closed-shell monomers and for density matrices obtained from the expectation-value expression with coupled cluster singles and doubles wave functions. The performance of the new approach has been demonstrated on several benchmark van der Waals systems, including dimers of argon, water, and ethyne.     

First-order exchange energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new method for the calculation of the first-order intermolecular exchange energy is proposed. It is based on the partition of two-particle density matrices of monomers into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. This partition is used to modify the formula for the first-order exchange energy developed by Moszynski et al. [J. Chem. Phys. 100, 5080 (1994)]. The new expression has been applied for the case of monomer density matrices derived from the expectation value expression for the coupled cluster singles and doubles wave function. In this way an accurate method of calculation of the first-order exchange energy for many-electron systems has been obtained, where both monomers are described on the coupled cluster level. Numerical results are presented for several benchmark van der Waals systems to illustrate the performance of the new approach.     

Traces of spin-adapted reduced density matrices

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ² and Ŝ_z operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.     

Permutational symmetry of reduced density matrices

The problems of permutational symmetry of the density matrices in reduction are studied. Some necessary and sufficient conditions for N, [λ₁], [λ₂]-derivability problem are given.     

Hierarchy equations for reduced density matrices in different representations

We derive exact relationships for the reduced density matrices in representations where the transformation matrix is a product of one-body transformation matrices. We specialize to the momentum and onebody energy representations. By decoupling the equations we are able to write the Hartree-Fock equation in terms of the first-order density matrix in an arbitrary representation. Applications to reduced local energy and the correlation problem are discussed.     

N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

Wick’s theorem and reconstruction schemes for reduced density matrices

We first obtained a closed form of the Wick’s theorem expressed in Grassman wedge product, which is similar to a binomial expansion. With this new expansion, new reconstruction schemes for reduced density matrices are derived rigorously. The higher order reduced density matrices are systematically decomposed into a sum of the lower order reduced density matrices which could be used to solve the contracted Schr?dinger equation.     

Structure and spin-purity conditions for reduced density matrices of arbitrary order

For arbitrary k, the separation of spin variables is performed in the reduced density matrix of the kth order (RDM -k) on the basis of the Fock coordinate function method. The independent spatial components of RDM -k are analyzed. For RDM -k of the total spin eigenstate, their number is proved never to exceed its spin multiplicity 2s + 1. Integral and other nontrivial interrelations between spatial components are established which turn out to be the necessary and sufficient conditions of spin purity of a wavefunction corresponding to a given RDM -k. It is shown that the r-rank k-particle spin distribution matrix F, defined as a spatial coefficient at the spin-tensorial operator of rank r in the RDM -k expansion, can be obtained by reduction of the (k + r)-particle charge density matrix F. In particular, all spatial components of RDM -2 are explicitly expressed in terms of the four-electron charge density matrix only. This allows us to purpose some approximative formulas for the McWeeny-Mizuno spin–orbit and spin–spin coupling functions in the case of the weak spin contamination.     

Description of functional groups by means of domain-restricted reduced density matrices

This report constitutes an application of our previous theoretical works on partitionings of the first-order reduced density matrix according to the atomic domains defined in the theory of atoms in molecules. The numerical determinations obtained reveal that the domain-restricted reduced density matrices, which are the tools resulting from the former treatments, are suitable devices to describe chemical features of molecular fragments. We have focused attention on a study of functional groups in several series of organic compounds confirming the usefulness of these tools.     

Contraction algorithms for third-order reduced density matrices: Symmetric group approach

In this work, we present the mapping formulae for the contraction of the third-order reduced density matrices represented in the basis of the irreducible representations (IR) of the symmetric group S3 into the second-order ones which are represented in the basis of the IR of group S2. These algorithms, which can be useful in several fields, have been applied for the approximation of reduced density matrices within the spin-adapted reduced Hamiltonian theory. Some results obtained with this procedure are also presented.     

Ensemble-representable reduced density matrices suggested by the Xα transition state

The transition state for the calculation of excitation energies in the Xα method is considered in terms of the exact reduced density matrices. It is shown that the occupation numbers which define the transition state correspond, in the exact case, not to a configuration interaction but to an ensemble of two single determinants.     

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇ V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.     

A study of the relationships between unpaired electron density, spin-density and cumulant matrices

This work describes the derivation of simple relationships between the density matrix of effectively unpaired electrons and the spin-density matrix in N-electron systems. The link between both devices turns out to be the one-electron matrix arising from the diagonal contraction of the cumulant matrix corresponding to the second-order reduced density matrix. We study some features of this contracted matrix, showing its usefulness to describe the electronic correlation. Numerical determinations performed in selected systems with different spin symmetries confirm the theoretical predictions.     

Some properties of reduced density matrices,correlated and uncorrelated,for pure and mixed states

Some results pertaining to the form of Γ⁽¹⁾ usually assumed in density functional theory, the properties of natural states of mixed states, their variational characterization, and a representation of pure states through reproducing kernels are given.     

Ab initio dynamics with wave-packets and density matrices

Efficient methodologies to conduct simultaneous dynamics of electrons and nuclei are discussed. Particularly, attention is directed to a recent development that combines quantum dynamics with ab initio molecular dynamics. The two components of the methodology, namely, quantum dynamics and ab initio molecular dynamics, are harnessed together using a time-dependent self-consistent field-like coupling procedure. An approach to conduct quantum dynamics using an accurate banded, sparse and Toeplitz representation for the discrete free propagator is highlighted with suitable review of other related approaches. One notable feature of the method is that all important quantum dynamical effects including zero-point effects, tunneling as well as over-barrier reflections are accurately treated. Computational methodologies for improved efficiency of the quantum dynamics are also discussed. There exists a number of ways to carry out simultaneous ab initio molecular dynamics (such as Born–Oppenheimer dynamics and extended Lagrangian dynamics, Car–Parrinello dynamics being a prime example of the latter); our prime focus remains on atom-centered density-matrix propagation and Born–Oppenheimer dynamics. The electronic degrees of freedom are handled at accurate levels of density functional theory, using hybrid or gradient corrected approximations. Benchmark calculations are provided for a prototypical proton transfer system. Future generalizations and goals are discussed.     

Two-electron reduced density matrices from the anti-Hermitian contracted Schrodinger equation: enhanced energies and properties with larger basis sets

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.