Orthogonal natural atomic orbitals form an appropriate one-electron basis for expanding CASSCF wave functions into localized bonding schemes and their weights

Localized bonding schemes and their weights have been obtained for the pi-electron system of nitrone by expanding complete active space self-consistent field wave functions into a set of Slater determinants composed of orthogonal natural atomic orbitals (NAOs) of Weinhold and Landis (Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, 2005). Thus, the derived bonding schemes are close to orthogonal valence bond structures. The calculated sequence of bonding scheme weights accords with the sequence of genuine resonance structure weights derived previously by Ohanessian and Hiberty (Chem Phys Lett 1987, 137, 437), who employed nonorthogonal atomic orbitals. This accord supports the notion that NAOs form an appropriate orthogonal one-electron basis for expanding complete active space self-consistent field wave functions into meaningful bonding schemes and their weights.     

On the rapid evaluation of cofactors in the calculation of nonorthogonal matrix elements

The calculation of matrix elements involving nonorthogonal orbitals is speeded up by recognizing the orthogonalities between orbitals, leading to generalized Slater rules. The block structure present in the overlap matrix makes an efficient evaluation of its cofactors possible. These cofactors are calculated per subblock, each with its own parity sign. An adjustment parity sign has to be evaluated, which is added to the combined local signs, to give the correct total sign for the matrix element. An algorithm for the evaluation of this adjustment sign has been developed, making an easy and correct evaluation possible. The current scheme is shown to be very efficient, but possibilities for further improvement remain. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 77–83, 1998     

Iterative diagonalization for orbital optimization in natural orbital functional theory

A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F . The off‐diagonal elements of the latter are determined explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be determined from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calculated with the Hartree‐Fock orbitals after the occupation numbers have been optimized. The method has been tested on the G2/97 set of molecules for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off‐diagonal elements of F . The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Calculation of molecular systems via coordinate wave functions

The system of charges is in a state with a given total spin S, which is described by a configuration of one-electron orbitals with arbitrary filling (subject to the Pauli principle). Expressions are derived for the matrix elements of operators F and G that are independent of the spin. The energy of the interaction between the completely filled orbitals and the singly filled ones is found to be independent of the spin of the latter. The formulas may be used with the tables of [2] to derive directly the expressions for the matrix elements of a configuration having an arbitrary number of completely filled orbitals and up to six singly filled ones.     

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries. Contribution of the Mark S. Gordon 65th Birthday Fegtschrift Issue.     

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.     

Second quantization for systems with a constant number of particles in the Dirac notation

The relation between the completeness condition for an appropriate one-particle basis set and the occupation number representation (second quantization) is shown for the time-independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one-electron density matrix and with the projector on the Fermi sea in the one-electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one-electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.     

Calculation of transition matrix elements by nonsingular orbital transformations

A general strategy is described for the evaluation of transition matrix elements between pairs of full class CI wave functions built up from mutually nonorthogonal molecular orbitals. A new method is proposed for the counter‐transformation of the linear expansion coefficients of a full CI wave function under a nonsingular transformation of the molecular‐orbital basis. The method, which consists in a straightforward application of the Cauchy–Binet formula to the definition of a Slater determinant, is shown to be simple and suitable for efficient implementation on current high‐performance computers. The new method appears mainly beneficial to the calculation of miscellaneous transition matrix elements among individually optimized CASSCF states and to the re‐evaluation of the CASCI expansion coefficients in Slater‐determinant bases formed from arbitrarily rotated (e.g., localized or, conversely, delocalized) active molecular orbitals. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Extracting covalent and ionic structures from usual delocalized wave functions: the electron-expansion methodology

We present easily programmable expansions, allowing the calculation of the weights of local covalent and ionic structures of a chemical bond from usual delocalized wave functions; they are obtained in the framework of the electron-expansion methodology, in which the hole conditions (involved by definition in a covalent or ionic structure) are expanded in terms involving only electrons. From the derived relations, true for both HF and correlated levels, one can also express the covalency/ionicity and the localization of a usual two-electron two-center (2e/2c) bond in terms of electronic populations. The three-electron populations are crucial for bond localization. On the contrary, in 2e/2c bonding, and particularly in Charge-Shift bonds (which show enhanced covalent-ionic interactions) although the three-electron populations can be non-negligible, they are not important for the covalency/ionicity of these bonds. Numerical applications and discussion are given for correlated MO wave functions of butadiene, hexatriene, and pyrrole molecules on the basis of both natural atomic orbitals (NAOs) (orthogonal orbitals) and pre-NAOs (nonorthogonal orbitals).     

Multiconfiguration self-consistent field,theory of localized orbitals: Choice of localization potential

We investigate the properties of two different choices for localization potentials for the direct construction of localized fixed orbitals by multiconfiguration self-consistent field theory. The first potential yields maximally screened orbitals by solution of a complicated orbital equation which depends explicitly on the complete set of orbitals for the system, and contains both one-and two-center matrix elements. The second localization potential yields somewhat less well screened orbitals by solution of a considerably simpler orbital equation which only contains simple one-center matrix elements.     

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.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

Valence and extra-valence orbitals in main group and transition metal bonding

We address the issue first raised by Maseras and Morokuma with regard to the questionable treatment of empty p-orbitals in the algorithm for natural atomic/bond orbitals (NAOs, NBOs) and associated natural population analysis. We quantify this issue in terms of the numerical error (root-mean-square density deviation) resulting from the two alternative treatments of empty p-sets, leading to distinct NAOs, atomic charges, and idealized Lewis structural representations. Computational application of this criterion to a broad spectrum of main group and transition group species (employing both single- and multi-structure resonance models) reveals the interesting general pattern of (i) relatively insignificant differences for normal-valent species, where a single resonance structure is usually adequate, but (ii) clear superiority of the standard NAO algorithm for hypervalent species, where multi-resonance character is pronounced. These comparisons show how the divisive issue of "valence shell expansion" in transition metal bonding is deeply linked to competing conceptual models of hypervalency (viz., "p-orbital participation" in skeletal hybridization vs. 3c/4e resonance character). The results provide a quantitative measure of superiority both for the standard NAO evaluation of atomic charges as well as the general 3c/4e (A: B-C<-->A-B :C resonance) picture of main- and transition-group hypervalency.     

On the transferability of Fock matrix elements

The transferability of Fock matrix elements in the linear combination of atomic orbitals molecular orbital scheme is analysed using localized orbitals. It is shown that this transferability is dependent on the transferability of these localized orbitals and the neglect of long-range contributions from partially cancelling Coulomb nuclear attraction and electron repulsion terms. A theoretical basis is thus provided for the simulated ab initio molecular orbital and related methods. Various corrections previously introduced in an ad hoc manner are shown to be justified. Transferability in both the closed shell and open shell schemes is analysed.     

Quasi-one-dimensional systems and localized defects: Some results of the Green's matrix formalism

The application of the self-consistent field (SCF ) local-impurity formalism to quasi-one-dimensional systems is discussed. We describe a general procedure for an accurate numerical determination of the Green's function matrix elements of the unperturbed system. An application to a local impurity in a model chain with two orbitals per unit cell is reported. The changes in the charge-bond-order matrix and in local and total density of states due to the impurity are discussed with special emphasis on the changes at the critical points (van Hove singularities) at the band edges. The Green's matrix approach is used to reexamine long-range Friedel oscillations caused by an impurity in a strictly one-dimensional metal. The extent of the long-range tail of the perturbed charge density is in an inverse relation to the localization length of the impurity state: the stronger the perturbation the more localized is the bound state and the more extended are the oscillations in the charge distribution. The results for the model chain with two orbitals per unit cell indicate that the impurity-induced change in charge distribution may be locally screened by redistribution of the population of the on-site orbitals, therefore damping possible oscillations and leading to a faster decay than in strictly one-dimensional systems.     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

Implementation of a density functional theory-based method for the calculation of the hyperfine A-tensor in periodic systems with the use of numerical and Slater type atomic orbitals: application to paramagnetic defects

The A-tensor parameterizes the "hyperfine" interaction of an "effective" electronic spin with the magnetic field due to the nuclear spin as monitored in an electron paramagnetic resonance (EPR) experiment. In this account, we describe an implementation for the calculation of the A-tensor in systems with translational invariance based on the Kohn-Sham form of density functional theory (KS DFT). The method is implemented in the periodic program BAND, where the Bloch states are expanded in the basis of numerical and Slater-type atomic orbitals (NAOs/STOs). This basis is well-suited for the accurate representation of the electron density near the nuclei, a prerequisite for the calculation of highly accurate hyperfine parameters. Our implementation does not rely on the frozen core approximation tacitly assumed in the pseudopotential schemes. The implementation is validated by performing calculations on the A-tensor for small atoms and molecules within the supercell approach as well as for paramagnetic defects in solids. In particular, we consider the A-tensor of "normal" and "anomalous" muonium defects in diamond and of the hydrogen cyanide anion radical HCN(-) in a KCl host crystal lattice.     

Lagrange function method for energy optimization directly in the space of natural orbitals

We propose a Lagrange method to obtain the electronic energy directly in the space of natural orbitals. The Lagrange function contains constraints to force the off‐diagonal elements of the one‐particle density matrix to zero, so the molecular orbitals converge to the natural orbitals simultaneously while the energy minimizes. The recently discovered “generalized Pauli conditions” for the occupation numbers are invoked to generate an initial approximation. As a demonstration of this approach, we study the system of three electrons in six orbitals which has become a paradigm for investigating multi‐particle entanglement.     

Relativistic correlation effects on the hyperfine structure and electric dipole transitions in Cs and Tl

A discrete numerical basis set is a versatile tool for many-body calculations. Here it is used to calculate second-order energy corrections and to construct approximate Brueckner orbitals for Cs and Tl in a relativistic framework. These orbitals, which often account for a large part of the correlation effects, are then used to evaluate the hyperfine structure and electric dipole transition matrix elements for a few low-lying states. The correlation effects were combined with the RPA diagrams, which account for the response of the orbitals to the external perturbation, and the results are compared with other calculations and with experiment. For Cs, the results are in good agreement with earlier work, whereas for the more complicated system Tl we find significantly larger contributions from the modification of the valence orbitals to approximate Brueckner orbitals.