Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

Spin-coupled theory for 'N electrons in M orbitals' active spaces

Spin-coupled (SC) theory, an ab initio valence bond (VB) approach which uses a compact and an easy-to-interpret single-orbital product wave function comparable in quality to a ‘N in N’ complete-active-space self-consistent field [CASSCF(N,N)] construction, is extended to ‘N in M’ (N ≠ M) active spaces. The SC(N,M) wave function retains the essential features of the original SC model: It involves just the products of nonorthogonal orbitals covering all distributions of N electrons between M orbitals in which as few orbitals as possible, |N – M|, are doubly occupied (for N > M) or missing (for N < M) and all other orbitals are singly occupied; each of these products is combined with a flexible spin function which allows any mode of coupling of the spins of the orbitals within the product. The SC(N,M) wave function remains much more compact than a CASSCF(N,M) construction; for example, the SC(6,7) wave function includes 35 configuration state functions (CSFs) as opposed to the 490 CSFs in the CASSCF case. The essential features of the SC(N,M) method are illustrated through a SC(6,5) calculation on the cyclopentadienyl anion, C5H5(–), and a SC(6,7) calculation on the tropylium cation, C7H7(+). The SC(6,5) and SC(6,7) wave functions for C5H5(–) and C7H7(+) are shown to provide remarkably clear modern VB models for the electronic structures of these aromatic cyclic ions which closely resemble the well-known SC model of benzene and yet recover almost all of the correlation energy included in the corresponding CASSCF(6,5) and CASSCF(6,7) wave functions: over 97% in the case of C5H5(–) and over 95% in the case of C7H7(+).     

Direct energy functional minimization under orthogonality constraints

The direct energy functional minimization problem in electronic structure theory, where the single-particle orbitals are optimized under the constraint of orthogonality, is explored. We present an orbital transformation based on an efficient expansion of the inverse factorization of the overlap matrix that keeps orbitals orthonormal. The orbital transformation maps the orthogonality constrained energy functional to an approximate unconstrained functional, which is correct to some order in a neighborhood of an orthogonal but approximate solution. A conjugate gradient scheme can then be used to find the ground state orbitals from the minimization of a sequence of transformed unconstrained electronic energy functionals. The technique provides an efficient, robust, and numerically stable approach to direct total energy minimization in first principles electronic structure theory based on tight-binding, Hartree-Fock, or density functional theory. For sparse problems, where both the orbitals and the effective single-particle Hamiltonians have sparse matrix representations, the effort scales linearly with the number of basis functions N in each iteration. For problems where only the overlap and Hamiltonian matrices are sparse the computational cost scales as O(M2N), where M is the number of occupied orbitals. We report a single point density functional energy calculation of a DNA decamer hydrated with 4003 water molecules under periodic boundary conditions. The DNA fragment containing a cis-syn thymine dimer is composed of 634 atoms and the whole system contains a total of 12,661 atoms and 103,333 spherical Gaussian basis functions.     

Normalized irreducible tensorial matrix expansions applied to an effective sp-type Hamiltonian for the PES of water

Any matrix can be expanded on a basis of SU(2) normalized irreducible tensorial matrices, NITM , defined in terms of 3-j symbols or coupling coefficients of SU (2). The NITM transform under rotations according to Wigner's matrices. If one dimension of an NITM is odd and the other even, the tensor has half-integer rank. A simple NITM basis consists of all NITM having the same numbers of rows and columns as the expanded matrix. A compound NITM basis consists of two or more simple bases, each spanning a corresponding block in the expanded matrix. The choice of NITM basis for expanding an effective Hamiltonian matrix is a crucial step in formulating a model. To illustrate the use of a compound NITM basis, including nonsquare NITM , an effective sp-type overlap-free superposition Hamiltonian is constructed and applied to the photoelectron ionization potential spectrum of water.     

Variational treatment of the vibrational Hamiltonian for NH(3) and H(2)NO

The full vibrational Hamiltonian for the inversion of NH(3) and H(2)NO has been diagonalized in a basis set that is the direct product of functions of the inversion coordinate and of harmonic vibrational functions independent of this inversion coordinate. The kinetic part of the Hamiltonian matrix is constructed with the use of the closure relation for these vibrational functions. The method is tested with the potential function which is supposed to be harmonic for the vibrations orthogonal to the inversion coordinate: the first computed levels are in good agreement with experimental levels for NH(3). For higher levels, anharmonic terms should be included.     

Analysis on the contribution of molecular orbitals to the conductance of molecular electronic devices

We present a theoretical approach which allows one to extract the orbital contribution to the conductance of molecular electronic devices. This is achieved by calculating the scattering wave functions after the Hamiltonian matrix of the extended molecule is obtained from a self-consistent calculation that combines the nonequilibrium Green's function formalism with density functional theory employing a finite basis of local atomic orbitals. As an example, the contribution of molecular orbitals to the conductance of a model system consisting of a 4,4-bipyridine molecule connected to two semi-infinite gold monatomic chains is explored, illustrating the capability of our approach.     

Franck-Condon factors based on anharmonic vibrational wave functions of polyatomic molecules

Franck-Condon (FC) integrals of polyatomic molecules are computed on the basis of vibrational self-consistent-field (VSCF) or configuration-interaction (VCI) calculations capable of including vibrational anharmonicity to any desired extent (within certain molecular size limits). The anharmonic vibrational wave functions of the initial and final states are expanded unambiguously by harmonic oscillator basis functions of normal coordinates of the respective electronic states. The anharmonic FC integrals are then obtained as linear combinations of harmonic counterparts, which can, in turn, be evaluated by established techniques taking account of the Duschinsky rotations, geometry displacements, and frequency changes. Alternatively, anharmonic wave functions of both states are expanded by basis functions of just one electronic state, permitting the FC integral to be evaluated directly by the Gauss-Hermite quadrature used in the VSCF and VCI steps [Bowman et al., Mol. Phys. 104, 33 (2006)]. These methods in conjunction with the VCI and coupled-cluster with singles, doubles, and perturbative triples [CCSD(T)] method have predicted the peak positions and intensities of the vibrational manifold in the X 2B1 photoelectron band of H2O with quantitative accuracy. It has revealed that two weakly visible peaks are the result of intensity borrowing from nearby states through anharmonic couplings, an effect explained qualitatively by VSCF and quantitatively by VCI, but not by the harmonic approximation. The X 2B2 photoelectron band of H2CO is less accurately reproduced by this method, likely because of the inability of CCSD(T)/cc-pVTZ to describe the potential energy surface of open-shell H2CO+ with the same high accuracy as in H2O+.     

改进的三维分离变量表象方法计算振动光谱

The modified discrete variable representation for three-dimension (DVR3D) method was applied to the determination of the vibrational energy levels of the fundamental electronic state of H2S and H2O. The Hamiltonian was expressed in Jacobi coordinates and developed on a DVR basis for each internal coordinate. The angular coordinate used a DVR based on Legendre polynomials and the radial coordinates utilized a DVR based on sine basis functions. Successive diagonalization and truncation technique was used to reduce the size of the final Hamiltonian matrix to be diagonalized. Calculations were presented for H2S and H2O to demonstrate the accuracy of these algorithms.     

On the vibronic coupling approximation: a generally applicable approach for determining fully quadratic quasidiabatic coupled electronic state Hamiltonians

In this report we introduce an iterative procedure for constructing a quasidiabatic Hamiltonian representing N(state)-coupled electronic states in the vicinity of an arbitrary point in N(int)-dimensional nuclear coordinate space. The Hamiltonian, which is designed to compute vibronic spectra employing the multimode vibronic coupling approximation, includes all linear terms which are determined exactly using analytic gradient techniques. In addition, all [N(state)][N(int)] quadratic terms, where [n]=n(n+1)/2, are determined from energy gradient and derivative coupling information obtained from reliable multireference configuration interaction wave functions. The use of energy gradient and derivative coupling information enables the large number of second order parameters to be determined employing ab initio data computed at a limited number of points (N(int) being minimal) and assures a maximal degree of quasidiabaticity. Numerical examples are given in which quasidiabatic Hamiltonians centered around three points on the C(3)H(3)N(2) potential energy surface (the minimum energy point on the ground state surface and the minimum energy points on the two- and three-state seams of conical intersection) were computed and compared. A method to modify the conical intersection based Hamiltonians to better describe the region of the ground state minimum is introduced, yielding improved agreement with ab initio results, particularly in the case of the Hamiltonian defined at the two-state minimum energy crossing.     

Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix

Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Pade? spectrum decomposition. This Lorentzian-Pade? decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Pade? decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.     

A general nonlinear expansion form for electronic wave functions

A new expansion form is presented for electronic wave functions. The wave function is a linear combination of product basis functions, and each product basis function in turn is formally equivalent to a linear combination of configuration state functions that comprise an underlying linear expansion space. The expansion coefficients that define the basis functions are nonlinear functions of a smaller number of variables. The expansion form is appropriate for both ground and excited states and to both closed and open shell molecules. The method is formulated in terms of spin-eigenfunctions using the graphical unitary group approach (GUGA), and consequently it does not suffer from spin contamination.     

Linear scaling density matrix perturbation theory for basis-set-dependent quantum response calculations: an orthogonal formulation

Linear scaling density matrix perturbation theory [A. M. N. Niklasson and M. Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is extended to basis-set-dependent quantum response calculations for a nonorthogonal basis set representation. The generalization is achieved by a perturbation-dependent congruence transform, derived from the factorization of the inverse overlap matrix, which transforms the generalized eigenvalue problem to an orthogonal, standard form. With this orthogonalization transform the basis-set-dependent perturbation in the overlap matrix is included in the orthogonalized Hamiltonian, which is expanded in orders of the perturbation. In this way density matrix perturbation theory developed for an orthogonal representation can be applied also to basis-set-dependent response calculations. The method offers an alternative to the previous solution of the basis-set-dependent response problem, based on a nonorthogonal generalization of the density matrix perturbation theory, where the calculations are performed within a purely nonorthogonal setting [A. M. N. Niklasson et al., J. Chem. Phys. 123, 44107 (2005)].     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

A diagrammatic valence bond method for configuration interaction calculations in atoms and molecules

A diagrammatic valence bond method based on Rumer-Pauling rules for configuration interaction calculations is described. The advantages of this method are that it is simple and flexible and is expected to be computationally efficient as the basis functions can be coded as increasing integers. Evaluation of Hamiltonian matrix elements involves simple bit manipulations and binary searches. The basis, being represented pictorially, should also help in utilizing spatial symmetries for further block-diagonalizing the Hamiltonian matrix. The eigenfunctions of the Hamiltonian can also be used to compute matrix elements between different electronic states.     

Localized impurity states in the Hartree-Fock,LCAO approximation. I.

One-electron energies and wave functions for deep trap impurity electrons in a crystal are calculated by the Hartree-Fock, single determinant method. The interactions arising from a many-electron single determinant crystal wave function, with automatic inclusion of exchange effects, are those which determine the one-electron functions and energies. The crystal plus impurity system has no translational symmetry and hence the Bloch theorem is not applicable for the solution of the essentially infinite Hartree-Fock eigenvalue matrix. Thus we develop a technique in which the Hamiltonian and overlap matrices are written in terms of bordered matrices, with the interaction of the impurity functions with the rest of the crystal environment contained in the bordering rows and columns. The resulting secular equation explicitly includes the effects of orthogonalization of the entire basis set, including the impurity functions. This technique could be used in an iterative calculation of the electronic structure of a small number of electrons, assuming that the rest of the electrons in the environment are fixed according to an initial estimate.     

The Sigma(-) states of the molecular hydrogen

A class of doubly excited electronic states of the hydrogen molecule is reported. The states are of Sigma(-) symmetry and are located ca. 200,000 cm(-1) above the ground state and about 75,000 cm(-1) above the ionization threshold. The electronic wave functions employed to described these states have been expanded in the basis of exponentially correlated Gaussian (ECG) functions with the nonlinear parameters variationally optimized. The lowest (3)Sigma and (1)Sigma states dissociate into hydrogen atoms in the n = 2 state, whereas the lowest (3)Sigma and (1)Sigma states have H(n = 2) and H(n = 3) as the dissociation products. All the four states are attractive and accommodate vibrational levels. The location of the vibrational energy levels has been determined by solving the radial Schr?dinger equation within the Born-Oppenheimer approximation.     

An application of perturbation theory ideas in configuration interaction calculations

Configuration interaction calculations of electronic wave functions for atoms and molecules have generally been limited to relatively small basis sets because of the exponential increase in the number of configurations as basis functions are added. While higher than quadruply excited configurations are of negligible importance in CI wave functions, it is shown that the effect of triple and quadruple excitation configurations can be substantially included even when the matrix elements between such configurations are neglected, leaving only their diagonal elements and the elements connecting them with the single and double excitations. This approximation is seen to be formally practically equivalent to a first-order perturbation expression for the wave function (second-order for the energy) based on an optimum linear combination of the zero, single, and double excitation configurations as the zero-order function. If suitable procedures are used, the amount of computational effort involved in such a calculation is roughly proportional to the fourth power of the number of basis functions employed, thus preventing the CI stage of the calculation from increasing in magnitude much faster than the stages involving the calculation and manipulation of the elementary integrals.     

Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory

The nuclear orbital plus molecular orbital (NOMO) theory was developed in order to determine the nonadiabatic nuclear and electronic wave functions. This study presents a formulation to remove the contamination of rotational motion as well as translational motion in the NOMO theory. We have formulated the translation- and rotation-free (TRF)-NOMO theory by introducing the TRF Hamiltonian. The principal moment of inertia, which is the denominator in the rotational Hamiltonian, is expanded in a Taylor series. The zeroth-order of the Taylor expansion corresponds to a rigid-body rotator. The first-order terms contribute the coupling between the vibration and the rotation. Hartree-Fock equations have been derived in the framework of the TRF-NOMO theory. Numerical assessments, which were preformed for H2, D2, T2, mu2 (muon dimmer), and H2O, confirmed the importance of the TRF treatment.     

Transformation of Bloch representation into atomic orbital representation and combined matrix element calculation technique for spatially bounded basis functions in crystals

A method is proposed for transforming the Hamiltonian from Bloch to atomic function representation. For spatially bounded functions, this is a rigorous method based on solution of a certain algebraic system of equations. Unlike the conventional procedure based on integration over the Brillouin zone, the new method requires knowledge of the matrix elements of the Bloch representation only at several points of the Brillouin zone. The number of these points is determined by the trimming radius for the spatially bounded functions and by the lattice constant. The method can be used for calculating matrix elements in a basis of atomic functions and for reducing computations in matrix element calculations of the Bloch representation for procedures using numerical integration.