Direct MRCI method for the calculation of relativistic many-electron wavefunctions. I. General formalism

The formalism of a quasi- or full-relativistic multireference CI method has been developed and implemented. The scheme is appropriate for the calculation of molecular systems in which the relativistic effects are of the same order of magnitude as the correlation contributions. In this contribution some important symmetry aspects of a relativistic many-electron wave function are discussed and the consequences for the CI matrix structure are shown. An efficient CI strategy in the form of a direct CI is presented, which avoids the construction of the whole CI matrix. Based on a determinantal expansion of molecular spinor products, the individual one- and two-electron molecular integrals are processed, and the molecular symmetry is easily accounted for by a proper linear combination of Slater determinants in the CI starting vector. For an efficient CI organization some powerful techniques of the graphical unitary group approach have been transferred to the relativistic case.     

Accurate and efficient treatment of two-electron contributions in quasirelativistic high-order Douglas-Kroll density-functional calculations

Two-component quasirelativistic approaches are in principle capable of reproducing results from fully relativistic calculations based on the four-component Dirac equation (with fixed particle number). For one-electron systems, this also holds in practice, but in many-electron systems one has to transform the two-electron interaction, which is necessary because a picture change occurs when going from the Dirac equation to a two-component method. For one-electron properties, one can take full account of picture change in a manageable way, but for the electron interaction, this would spoil the computational advantages which are the main reason to perform quasirelativistic calculations. Exploiting those picture change effects are largest in the atomic cores, which in molecular applications do not differ too much from the cores of isolated neutral atoms, we propose an elegant, efficient, and accurate approximation to the two-electron picture change problem. The new approach, called the "model potential" approach because it makes use of atomic (four- and two-component) data to estimate picture change effects in molecules, shares with the nuclear-only approach that the Douglas-Kroll operator needs to be constructed only once (not in each self-consistent-field iteration) and that no time-consuming multicenter relativistic two-electron integrals need to be calculated. The new approach correctly describes the screening of both the nearest nucleus and distant nuclei, for the scalar-relativistic as well as the spin-orbit parts of the Hamiltonian. The approach is tested on atomic and molecular-orbital energies as well as spectroscopic constants of the lead dimer.     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Using many-particle basis sets for electronic energy calculations of molecules

A method of calculating the electronic energies of molecules using a many-particle basis set is proposed. In this case, the many-particle Schrödinger equation may be solved without resorting to the one-electron approximation. The results of the electronic energy calculations of some two-electron systems are in good agreement with experimental results.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

A new parametrizable model of molecular electronic structure

A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, the variational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecular integrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to get its matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization procedure--making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li-F, Na-Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promising for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost.     

Asymmetrical linear structures including three-electron hemibonds or other interactions in the (ABA)-type triatomic cations: Ne3+, (He-Ne-He)+, (Ar-Ne-Ar)+, (Ar-O-Ar)+, (He-O-He)+, and (Ar-He-Ar)+

By the counterpoise geometry optimization at the level of CCSD(T)aug-cc-pVDZ, the asymmetrical linear structures with all the real frequencies were obtained for the triatomic cations of (ABA)+ type: Ne3+, (He-Ne-He)+, (Ar-Ne-Ar)+, (Ar-He-Ar)+, (He-O-He)+, and (Ar-O-Ar)+. The validity of this optimization method is confirmed by comparing with the method of the potential-energy surface for the calculations of Ne3+ and (He-Ne-He)+. Using the molecular-orbital theory, it is found that the interaction within the triatomic cations is dominated by the contribution from the first two atoms while the contribution from the third atom is small. This result is justified as a direct consequence of forming an asymmetrical linear structure. Specifically, four types of interaction within the triatomic cations are identified: three-electron sigma-type hemibond, three-electron pi-type hemibond, two-electron sigma bond, and the attraction between cation and atoms. For Ne3+, (He-Ne-He)+, and (He-O-He)+ clusters, it is shown that the electron correlation effect supports the asymmetry.     

超球坐标下原子、分子体系的变分计算：—氦原子和氢负离子基态

应用超球会标表示氦原子和氢负离子的薛定谔方程，将二电子原子在三维空间中的运动转化为单电子原子在六维空间中受广义库仑力作用的运动，我们给出了六维空间广义角动量算符的本征值与本征函数，并以此本征函数微基构造超球波函数，得到超球径微分方程，以广义Laguerre 多项式表示超球径波函数，运用密度矩阵和线性变分法得到非正交基下超球径波函数满足的久期方程，最后求得能量和波函数，计算结果与精确的计算符合良好。     

A new theory for symmetry orbital and tensor (Ⅱ)——Symmetric reduction of molecular integrals and self-consistent field calculations

The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation scheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries.     

On the importance of excited state dynamic response electron correlation in polarizable embedding methods

We investigate the effect of including a dynamic reaction field at the lowest possible ab inito wave function level of theory, namely the Hartree‐Fock (HF) self‐consistent field level within the polarizable embedding (PE) formalism. We formulate HF based PE within the linear response theory picture leading to the PE–random‐phase approximation (PE–RPA) and bridge the expressions to a second‐order polarization propagator approximation (SOPPA) frame such that dynamic reaction field contributions are included at the RPA level in addition to the static response described at the SOPPA level but with HF induced dipole moments. We conduct calculations on para‐nitro‐aniline and para‐nitro‐phenolate using said model in addition to dynamic PE–RPA and PE–CAM–B3LYP. We compare the results to recently published PE–CCSD data and demonstrate how the cost effective SOPPA‐based model successfully recovers a great portion of the inherent PE–RPA error when the observable is the solvatochromic shift. We furthermore demonstrate that whenever the change in density resulting from the ground state‐excited state electronic transition in the solute is not associated with a significant change in the electric field, dynamic response contributions formulated at the HF level of theory manage to capture the majority of the system response originating from derivative densities. © 2012 Wiley Periodicals, Inc.     

A simple but fully nonlocal correction to the random phase approximation

The random phase approximation (RPA) stands on the top rung of the ladder of ground-state density functional approximations. The simple or direct RPA has been found to predict accurately many isoelectronic energy differences. A nonempirical local or semilocal correction to this direct RPA leaves isoelectronic energy differences almost unchanged, while improving total energies, ionization energies, etc., but fails to correct the RPA underestimation of molecular atomization energies. Direct RPA and its semilocal correction may miss part of the middle-range multicenter nonlocality of the correlation energy in a molecule. Here we propose a fully nonlocal, hybrid-functional-like addition to the semilocal correction. The added full nonlocality is important in molecules, but not in atoms. Under uniform-density scaling, this fully nonlocal correction scales like the second-order-exchange contribution to the correlation energy, an important part of the correction to direct RPA, and like the semilocal correction itself. For the atomization energies of ten molecules, and with the help of one fit parameter, it performs much better than the elaborate second-order screened exchange correction.     

Nonvariational time-dependent multiconfiguration self-consistent field equations for electronic dynamics in laser-driven molecules

A time-dependent multiconfiguration self-consistent field (TDMCSCF) scheme is developed to describe the time-resolved electron dynamics of a laser-driven many-electron atomic or molecular system, starting directly from the time-dependent Schrodinger equation for the system. This nonvariational formulation aims at the full exploitations of concepts, tools, and facilities of existing, well-developed quantum chemical MCSCF codes. The theory uses, in particular, a unitary representation of time-dependent configuration mixings and orbital transformations. Within a short-time, or adiabatic approximation, the TDMCSCF scheme amounts to a second-order split-operator algorithm involving generically the two noncommuting one-electron and two-electron parts of the time-dependent electronic Hamiltonian. We implement the scheme to calculate the laser-induced dynamics of the two-electron H2 molecule described within a minimal basis, and show how electron correlation is affected by the interaction of the molecule with a strong laser field.     

A new theory for symmetry orbital and tensor (II)

The symmetry orbital-symmetry orbital tensor methtd is applied to the evaluation of molecular integrals (one-electron) and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation sheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries. Projcct supported by the National Natural Science Foundation of China (Grant No. 29473119).     

Calculations of solute and solvent entropies from molecular dynamics simulations

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.     

Third-order corrections to random-phase approximation correlation energies

Several random-phase approximation (RPA) correlation methods were compared in third order of perturbation theory. While all of the considered approaches are exact in second order of perturbation theory, it is found that their corresponding third-order correlation energy contributions strongly differ from the exact third-order correlation energy contribution due to missing interactions of the particle-particle-hole-hole type. Thus a simple correction method is derived which makes the different RPA methods also exact to third-order of perturbation theory. By studying the reaction energies of 16 chemical reactions for 21 small organic molecules and intermolecular interaction energies of 23 intermolecular complexes comprising weakly bound and hydrogen-bridged systems, it is found that the third-order correlation energy correction considerably improves the accuracy of RPA methods if compared to coupled-cluster singles doubles with perturbative triples as a reference.     

The Physical Mechanism of the Chemical Bond

First the interplay of kinetic and potential energy via the uncertainty relation is described with the aid of a variational function for the ground state of the H atom. The H ion is used to illustrate the physical mechanism of the occurrence of the chemical bond. The formation of the chemical bond can be divided into three steps: 1. the quasi-classical (electrostatic) interaction of the unchanged electronic charges of the separate atoms; 2. the interference of the atomic orbitals, which (in the case of positive interference) leads to a displacement of charge into the bonding region and to a decrease in the kinetic energy; 3. deformation of the molecular orbitals to restore the correct balance of kinetic and potential energy. In simple models, it is often sufficient to consider just the second step. A two-electron bond is not fundamentally different from a one-electron bond. In larger molecules it is possible to distinguish between long-range and short-range interatomic contributions to the chemical bond. If the former are small, i. e. in molecules with non-polar bonds, a one-electron molecular orbital theory can be justified. Finally, the possibility of describing molecules by localized bonds is discussed.     

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.     

Linear-scaling formation of Kohn-Sham Hamiltonian: application to the calculation of excitation energies and polarizabilities of large molecular systems

We present calculations of excitation energies and polarizabilities in large molecular systems at the local-density and generalized-gradient approximation levels of density-functional theory (DFT). Our results are obtained using a linear-scaling DFT implementation in the program system DALTON for the formation of the Kohn-Sham Hamiltonian. For the Coulomb contribution, we introduce a modification of the fast multipole method to calculations over Gaussian charge distributions. It affords a simpler implementation than the original continuous fast multipole method by partitioning the electrostatic Coulomb interactions into "classical" and "nonclassical" terms which are explicitly evaluated by linear-scaling multipole techniques and a modified two-electron integral code, respectively. As an illustration of the code, we have studied the singlet and triplet excitation energies as well as the static and dynamic polarizabilities of polyethylenes, polyenes, polyynes, and graphite sheets with an emphasis on the trends observed with system size.