Hamiltonian for diabatic molecular states

Hyperradial-adiabatic potential curves for (dt )⁺ molecular ions calculated for states with total three-body angular momentumJ=1 and total parityp=1, i.e. for the abnormal parity ground set, are demonstrated to be exactly diabatic-hyperradial potential curves for (J=1,p=–1) symmetry adiabatic curves. This could be thought of as a formal manifestation of the -doubling effect. The projection operator method of constructing a diabatic Hamiltonian that was suggested earlier can be naturally used in our case, too.     

Perturbation theory corrections to the two-particle reduced density matrix variational method

In the variational 2-particle-reduced-density-matrix (2-RDM) method, the ground-state energy is minimized with respect to the 2-particle reduced density matrix, constrained by N-representability conditions. Consider the N-electron Hamiltonian H(lambda) as a function of the parameter lambda where we recover the Fock Hamiltonian at lambda=0 and we recover the fully correlated Hamiltonian at lambda=1. We explore using the accuracy of perturbation theory at small lambda to correct the 2-RDM variational energies at lambda=1 where the Hamiltonian represents correlated atoms and molecules. A key assumption in the correction is that the 2-RDM method will capture a fairly constant percentage of the correlation energy for lambda in (0,1] because the nonperturbative 2-RDM approach depends more significantly upon the nature rather than the strength of the two-body Hamiltonian interaction. For a variety of molecules we observe that this correction improves the 2-RDM energies in the equilibrium bonding region, while the 2-RDM energies at stretched or nearly dissociated geometries, already highly accurate, are not significantly changed. At equilibrium geometries the corrected 2-RDM energies are similar in accuracy to those from coupled-cluster singles and doubles (CCSD), but at nonequilibrium geometries the 2-RDM energies are often dramatically more accurate as shown in the bond stretching and dissociation data for water and nitrogen.     

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Mu?ller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.     

A spherically averaged representation of the atomic one-particle reduced density matrix

Summary A new way of representing the one-particle reduced density matrix (ODM) of closed-shell atoms in a spherically averaged manner is presented, and connections of this representation to the radial density distributionD(R) and the isotropic reciprocal form factorB(s) are shown. In this representation, certain characteristics of the angular nodal structure of the natural orbitals (NOs) are preserved. Examples of hydrogenic orbitals and near-Hartree-Fock wave functions for some closed-shell atoms are given.     

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.     

A fast fourier transform method for calculating the equilibrium density matrix

We calculate the low-temperature quantum density matrix by integrating numerically the Bloch equation. The initial condition is the classical high-temperature value of the density matrix. The integration method uses short “time” propagators computed by a fast Fourier transform method.     

A novel interpretation of reduced density matrix and cumulant for electronic structure theories

We propose a novel interpretation of the reduced density matrix (RDM) and its cumulant that combines linear and exponential parametrizations of the wavefunction. Any n-particle RDM can be written as a weighted average of "configuration interaction" amplitudes. The corresponding n-particle cumulant is represented in terms of two types of contributions: "connected" (statistical averages of substitution amplitudes) and "disconnected" (cross-correlations of substitution amplitudes). A diagonal element of n-RDM represents the average occupation number of the orbital n-tuple. The diagonal elements of 2- and 3-cumulants take particularly elegant forms in the natural spin-orbital basis: they represent the covariances (correlated fluctuations) of the occupation numbers of the orbital pair and triples, respectively. Thus, the diagonal elements of the cumulants quantify the correlation between the orbital occupation numbers. Our interpretation is used to examine the weak to strong correlation transition in the "two electrons in two orbitals" problem.     

Obtaining the two-body density matrix in the density matrix renormalization group method

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.     

Evaluation of the second-order reduced density matrix for correlated electronic wave functions

Formulas are presented for the evaluation of the second-order reduced density matrix for correlated wave functions, including wave functions constructed from pairwise nonorthogonal orbitals. Numerical results are provided for the correlation holes and conditional nuclear spin densities for the water molecule.     

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.     

Computation of dipole, quadrupole, and octupole surfaces from the variational two-electron reduced density matrix method

Recent advances in the direct determination of the two-electron reduced density matrix (2-RDM) by imposing known N-representability conditions have mostly focused on the accuracy of molecular potential energy surfaces where multireference effects are significant. While the norm of the 2-RDM's deviation from full configuration interaction has been computed, few properties have been carefully investigated as a function of molecular geometry. Here the dipole, quadrupole, and octupole moments are computed for a range of molecular geometries. The addition of Erdahl's T2 condition [Int. J. Quantum Chem. 13, 697 (1978)] to the D, Q, and G conditions produces dipole and multipole moments that agree with full configuration interaction in a double-zeta basis set at all internuclear distances.     

An efficient linear scaling procedure for constructing localized orbitals of large molecules based on the one-particle density matrix

We have developed a linear-scaling algorithm for obtaining the Boys localized molecular orbitals from the one-particle density matrix. The algorithm is made up of two steps: the Cholesky decomposition of the density matrix to obtain Cholesky molecular orbitals and the subsequent Boys localization process. Linear-scaling algorithms have been proposed to achieve linear-scaling calculations of these two steps, based on the sparse matrix technique and the locality of the Cholesky molecular orbitals. The present algorithm has been applied to compute the Boys localized orbitals in a number of systems including α-helix peptides, water clusters, and protein molecules. Illustrative calculations demonstrate that the computational time of obtaining Boys localized orbitals with the present algorithm is asymptotically linear with increasing the system size.     

The structure of the second-order reduced density matrix in density matrix functional theory and its construction from formal criteria

Some formal requirements for the second-order reduced density matrix are discussed in the context of density matrix functional theory. They serve as a basis for the ad hoc construction of the second-order reduced density matrix in terms of the first-order reduced density matrix and lead to implicit functionals where the occupation numbers of the natural orbitals are obtained as diagonal elements of an idempotent matrix the elements of which represent the variational parameters to be optimized. The numerical results obtained from a first realization of such an implicit density matrix functional give excellent agreement with the results of full configuration interaction calculations for four-electron systems like LiH and Be. Results for H2O taken as an example for a somewhat larger molecule are numerically less satisfactory but still give reasonable occupation numbers of the natural orbitals and indicate the capability of density matrix functional theory to cope with static electron correlation.     

Continuum limit frozen Gaussian approximation for the reduced thermal density matrix of dissipative systems

A continuum limit frozen Gaussian approximation is formulated for the reduced thermal density matrix for dissipative systems. The imaginary time dynamics is obtained from a novel generalized Langevin equation for the system coordinates. The method is applied to study the thermal density in a double well potential in the presence of Ohmic-like friction. We find that the approximation describes correctly the delocalization of the density due to quantization of the vibrations in the well. It also accounts for the friction induced reduction of the tunneling density in the barrier region.     

Perturbation theory for the density matrix in the MO LCAO method

The problem of calculating the density matrix of perturbed electron shells in complex molecules has been solved in the MO LCAO [molecular orbital linear combination atomic orbitals] approximation. Considering the electron interaction in correcting the density matrix in each order of the perturbation theory yields one matrix equation uniquely defining the correction. This equation is formulated as an ordinary, linear nonhomogeneous system with respect to the individual matrix elements; this provided the possibility of establishing the convergence region of the self-consistent method. The solution of this system is accomplished by the method of steepest descent which, as has been shown, is always convergent in this case. It was established that just as in the conventional perturbation theory, the (2i + 1)th correction for the energy in terms of the Hartree-Fock method may be expressed by a correction of the density matrix not exceeding the i-th order. Detailed expressions are presented up to the eighth energy correction.     

A new Monte Carlo method for getting the density of states of atomic cluster systems

A novel Monte Carlo flat histogram algorithm is proposed to get the classical density of states in terms of the potential energy, g(E(p)), for systems with continuous variables such as atomic clusters. It aims at avoiding the long iterative process of the Wang-Landau method and controlling carefully the convergence, but keeping the ability to overcome energy barriers. Our algorithm is based on a preliminary mapping in a series of points (called a σ-mapping), obtained by a two-parameter local probing of g(E(p)), and it converges in only two subsequent reweighting iterations on large intervals. The method is illustrated on the model system of a 432 atom cluster bound by a Rydberg type potential. Convergence properties are first examined in detail, particularly in the phase transition zone. We get g(E(p)) varying by a factor 10(3700) over the energy range [0.01 < E(p) < 6000 eV], covered by only eight overlapping intervals. Canonical quantities are derived, such as the internal energy U(T) and the heat capacity C(V)(T). This reveals the solid to liquid phase transition, lying in our conditions at the triple point. This phase transition is further studied by computing a Lindemann-Berry index, the atomic cluster density n(r), and the pressure, demonstrating the progressive surface melting at this triple point. Some limited results are also given for 1224 and 4044 atom clusters.     

The density matrix in the two-particle function method

The first- and second-order density matrices D ^(N) and D for the function g⁽ⁿ⁾ = A_N[g(1, 2) …? g(N ? 1, N)] are expressed by the g function itself and its density matrix D . In a singlet state the generating functions for spatial parts of these matrices are simply connected with there solvent of the Fredholm equation in which the spatial part of D is a kernel. Some special cases of g(1, 2) are considered. It isestablished that the number of large eigenvalues of D does not exceed that of different eigenvalues of D . Thus the degeneracy in the spectrum of D causes the appearance of such large eigenvalues.     

The density matrix in the unrestricted Hartree-Fock method

The Hartree-Fock approach is applied subject to orthogonality in the one-electron functions to obtain the two-particle density matrix for the pure spin state. Expressions are given for the two-particle spin density, the paired-spin density, and the anisotropy function of the pairing.     

Improved density of states Monte Carlo method based on recycling of rejected states

In this paper a new algorithm is presented that improves the efficiency of Wang and Landau algorithm or density of states (DOS) Monte Carlo simulations by employing rejected states. The algorithm is shown to have a performance superior to that of the original Wang-Landau [F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001)] algorithm and the more recent configurational temperature DOS algorithm. The performance of the method is illustrated in the context of results for the Lennard-Jones fluid.