Variational principles for discontinuous wave functions and the independent particle model of electronic structure

Using the variational method advanced by McCavert and Rudge, we obtain the independent particle model variational principle for loge localized discontinuous wave functions. The transformation of the variational expressions into matrix form when the loge localized discontinuous orbitals are expanded in finite basis sets is discussed. The simplifications brought about by this new method in the evaluation of molecular integrals are indicated.     

The electronic correlation problem and loge localization

The model of complete loge localization is employed here to develop a practical method for handling correlation effects in atomic and molecular many electron systems. Intraloge correlation is dealt with by an independent variational treatment of pair functions which are continuous and vanish outside a given loge. It is shown that in the context of the model it is possible to compute pair correlation energies for localized single and double bonds in molecules by evaluating only modified atomic integrals. We bypass in this manner the evaluation of multicenter integrals necessary in other formalisms. In addition, the corrections to the model are discussed and in particular it is shown that part of the interloge correlation effects are already described by the loge localized wave function.     

Self-consistent embedded clusters: Building block equations for localized orthogonal orbitals

In order to be able to study a large electronic system following a building block approach, in which smaller tractable subsystems are handled at a time rather than the system as a whole, equations are proposed in this paper whose solutions are variational orthogonal orbitals localized on the subsystems. The equations for a given subsystem correspond to a molecular cluster embedded in the field created by the rest of the system, and are coupled to the corresponding equations for all subsystems under consideration, so that they must be solved self-consistently. While the localized nature of the solutions makes the equations appropriate for use in conjunction with local basis sets in practical implementations without significant loss of precision due to truncation errors, their orthogonality properties allow for the use of the advantages of the theory of separability of McWeeny in order to calculate total energies and (generalized product) wave functions. Since the building block equations proposed involve inter-subsystem interactions very cumbersome to calculate, an approximation is proposed in order to make their application to actual problems feasible: the representation of the cumbersome interaction operators by ab initio model potentials which are obtained directly from them, without resorting to any parametrization procedure based on a reference. This ab initio model potential approximation has been found to provide considerable computational savings without significant loss of accuracy in frozen-core calculations on molecules and frozen-lattice calculations on imperfect crystals.     

Computing the energy of a water molecule using multideterminants: a simple, efficient algorithm

Quantum Monte Carlo (QMC) methods such as variational Monte Carlo and fixed node diffusion Monte Carlo depend heavily on the quality of the trial wave function. Although Slater-Jastrow wave functions are the most commonly used variational ansatz in electronic structure, more sophisticated wave functions are critical to ascertaining new physics. One such wave function is the multi-Slater-Jastrow wave function which consists of a Jastrow function multiplied by the sum of Slater determinants. In this paper we describe a method for working with these wave functions in QMC codes that is easy to implement, efficient both in computational speed as well as memory, and easily parallelized. The computational cost scales quadratically with particle number making this scaling no worse than the single determinant case and linear with the total number of excitations. Additionally, we implement this method and use it to compute the ground state energy of a water molecule.     

Molecular MC–SCF calculations

A practical method for finding multi-configurational SCF wave functions is proposed. The basic equation is equivalent to the Brillouin theorem; comparison with the usual SCF equations obtained through effective hamiltonians gives an interpretation of the offdiagonal Lagrange multipliers. Numerical applications to Formaldehyde in a minimum Slater-type orbital basis with four different variational wave functions are reported. The molecular orbitals found in these calculations are localized on the chemical bonds. The largest contributions to the energy are obtained from π-π and dispersion-type σ-π correlation.     

Natural J-coupling analysis: interpretation of scalar J-couplings in terms of natural bond orbitals.

The natural J-coupling (NJC) method presented here analyzes the Fermi contact portion of J-coupling in the framework of finite perturbation theory applied to ab initio/density function theory (DFT) wave functions, to compute individual and pairwise orbital contributions to the net J-coupling. The approach is based on the concepts and formalisms of natural bond orbital (NBO) methods. Computed coupling contributions can be classified as Lewis (individual orbital contributions corresponding to the natural Lewis structure of the molecule), delocalization (resulting from pairwise donor-acceptor interactions), and residual repolarization (corresponding to correlation-like interactions). This approach is illustrated by an analysis of the angular and distance dependences of the contributions to vicinal (3)J(HH) couplings in ethane and to the long-range (6)J(HH) couplings in pentane. The results indicate that approximately 70% or more of the net J-coupling is propagated by steric exchange antisymmetry interactions between Lewis orbitals (predominantly sigma bonding orbitals). Hyperconjugative sigma to sigma delocalization interactions account for the remainder of the coupling. Calculated pairwise-steric and hyperconjugative-delocalization energies provide a means for relating coupling mechanisms to molecular energetics. In this way, J-coupling contributions can be related directly to the localized features of the molecular electronic structure in order to explain measured J-coupling patterns and to predict J-coupling trends that have yet to be measured.     

Localization measure and maximum delocalization in molecular systems

A mathematically well-defined measure of localization is presented based on Mulliken's orbital populations. It is shown that this quantity equals 1 for core- and lone-pair orbitals, 2 for two-atomic bonds, 6 for benzene rings, etc., and it is applicable for delocalized canonical HF orbitals as well. The definition of this quantity is general in the sense that ab initio MOS with overlapping AO expansion, and semiempirical wave functions using the ZDO approximation as well, can be treated. The localization quantity is essentially “intrinsic,” i.e., no subdivision of the molecule is required. For N-electron wave functions, mean delocalization can be defined. This measure is not invariant to unitary transformations of the one-electron orbitals, characterizing in this way the localized or extended representation of the N-electron wave function. It can be proven, however, that for unitary transformed wave functions a maximum delocalization exists which depends only on the physical (N-electron) properties of the molecule. It is shown that inhomogeneous charge distribution can cause strong electron localization in molecular systems. The delocalization of the canonical Hartree–Fock orbitals, the Parr–Chen circulant orbitals, and the optimum delocalized orbitals is studied by numerical calculations in extended systems.     

Coordinate measurements and operators

Relativistic quantum-field theory provides the machinery for calculating wave functions or probability amplitudes depending upon space-time coordinates. The currently accepted theory, however, fails to provide position operators and a means of measuring particle coordinates that are consistent with Dirac's properties of physical observables. This is because it calls for a space position probability distribution at a specified time. This paper shows, however, that space-time event coordinate operators, together with a corresponding measurement procedure, can be found that are consistent with Dirac's requirements. This is done through a reinterpretation of the amplitudes computed by field theory and does not involve any change in that mathematical formalism. The measurement of the space-time coordinates of an event is accomplished by detecting the absorption of a photon by a particle from each of two light pulses designed to overlap at a given point at a given time. If a final emitted photon has an energy whose sum with the final particle energy approximately equals the sum of the mean energies of the pulses, then the absorption of the two pulse photons must certainly have taken place within a distance the order of a Compton wavelength of the small space-time region of overlapping pulses. This is clear from the fact that the high energy required to confine the pulses to very small volumes must throw a particle absorbing them far off the mass shell. Thus the absorption of the two photons throws the particle into a narrowly confined spatial wave function that must decay extremely rapidly—to within a Compton wavelength, a delta function in space-time. This delta function is the eigenfunction of space-time coordinate operators X^μ and is the scalar product of vectors in a Hilbert space spanned by spin–space-time kets large enough to contain the operators of the Poincaré group. These event operators transform properly under the action of Poincaré operators but do not commute with the mass. If the Compton wavelength is not negligible compared to the accuracy desired in the coordinate measurements, individual coordinate measurements are no longer possible. Nevertheless, a large number of repeated coordinate measurements can be carried out to produce a coordinate probability distribution. This distribution can be unfolded to find a true coordinate probability distribution if the charge form factor is known from basic theory. An analysis of laboratory particle detection techniques shows that they actually determine space coordinates and energy rather than spatial coordinates at a given time. When this fact is included, the Klein–Nishina formula can be derived using the electromagnetic four-vector potential as the photon probability amplitude wave. To clarify the meaning of the observables, a mass-momentum measurement is described.     

The use of explicitly correlated,partially antisymmetric wave functions in atomic and molecular calculations

Trial wave functions, written as the sum of a configuration interaction expansion and an explicitly correlated term which is not antisymmetric, are proposed for use in calculating the electronic properties of atoms and molecules. A variational principle, modified to allow the use for such partially antisymmetric wave functions, is developed. It is shown that the consequences of partial antisymmetry on calculated expectation values can be estimated. The method avoids difficult three-electron integrals which arise in other theories.     

New bosonic excitation operators in many-electron wave functions

New excitation operators which have perfect bosonic symmetry are constructed for many-electron wave functions by regarding the system of many electrons as that of many species of bosons. Any electronic configurations can be generated by the new bosonic “void” operators. A coherent state is constructed with the bosonic operators and is adopted as a trial function for the time-dependent variational principle. The equation of motion which has exactly the same form as Hamilton's equation in classical mechanics is obtained with the complex variational parameters, the number of which is equal to the number of electrons. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 71–75, 1998     

Methods of composite molecular wave functions. I. Variational principle and multiconfiguration SCF theory

The Silverstone–Stuebing variational principle for the discontinuous wave functions of one-electron systems is generalized for many-electron systems. The variational functional of energy takes real or complex value. The condition that it is real is given. Using the generalized variational principle, a multiconfiguration SCF theory for the composite molecular wave function is formulated. According to the theory, we may divide the whole space into space-filling cells, solve the SCF equations in each cell and build up the wave functions of the system by gathering the wave functions obtained in the cells. For use in the basis-set expansion method, the SCF equations are rewritten as matrix forms in which only one- and two-center integrals appear if an expansion center is located in each cell.     

DFT approach to calculate electronic transfer through a segment of DNA double helix

The electronic structures of an entire segment of a DNA molecule were calculated in its single‐strand and double‐helix cases using the DFT method with an overlapping dimer approximation and negative factor counting method. The hopping conductivity of the segment was calculated by the random walk theory from the results of energy levels and wave functions obtained. The results of the single‐strand case show that the DFT method is quantitatively in agreement with that of the HF MP2 method. The results for the double helix are in good agreement with that of the experimental data. Therefore, the long‐range electron transfer through the DNA molecule should be caused by hopping of electronic charge carriers among different energy levels whose corresponding wave functions are localized at different bases of the DNA molecule. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1109–1117, 2000     

Excited states of boron isoelectronic series from explicitly correlated wave functions

The ground state and some low-lying excited states arising from the 1s2 2s2p2 configuration of the boron isoelectronic series are studied starting from explicitly correlated multideterminant wave functions. One- and two-body densities in position space have been calculated and different expectation values such as , , , , , and , where r, r12, and R stand for the electron-nucleus, interelectronic, and two electron center of mass coordinates, respectively, have been obtained. The energetic ordering of the excited states and the fulfillment of the Hund's rules is analyzed systematically along the isoelectronic series in terms of the electron-electron and electron-nucleus potential energies. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding noncorrelated ones. All the calculations have been done by using the variational Monte Carlo method.     

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field and breathing-orbital valence bond wave functions. Complemented by a Jastrow factor, these Jastrow-valence-bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C(2), N(2), O(2), and F(2) in both variational Monte Carlo and diffusion Monte Carlo. We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlations. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.     

Orbital-orthogonality constraints and basis-set optimization

A new procedure is presented for introducing arbitrary orbital-orthogonality constraints in the variational optimization of otherwise nonorthogonal multiconfiguration electronic wave functions. It is based on suitable analytical changes to the expressions for the first and second derivatives of the electronic energy with respect to the independent variational parameters, and can be applied in the presence of symmetry constraints. It is tested using a second-derivative optimization procedure, the Optimized Basis Set -- Generalized Multiconfiguration Spin-Coupled (OBS-GMCSC) approach, that can treat basis-function exponential parameters as variational parameters, to be optimized simultaneously with configuration, spin-coupling, and orbital coefficients. This enables rigorous optimization of basis-set exponential parameters even for fully orthogonal multiconfiguration wave functions. Test calculations are carried out, with optimized even-tempered basis sets, on Li(2) and on the CH radical. For the latter, special attention is paid to the electronic spin density at the nuclei.     

Correlated wave-function theory for many-center many-electron problems

The correlated electronic wave-function theory developed by S. Obara and K. Hirao [Bull. Chem. Soc. Jpn. 66 , 3300 (1993)], as applied to two-electron molecular systems, is generalized to many-center many-electron problems. The exact formulas for effective Hamiltonian operators are given. The rules for the calculation of matrix elements with three-electron operators over Slater determinants are formulated. From the energy-minimum principle, the system of master equations is derived for variational coefficients of a trial wave function for the molecules with closed electronic shells. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 639–648, 1998     

Simultaneous analytical optimization of variational parameters in Gaussian-type functions with full configuration interaction of multicomponent molecular orbital method by elimination of translational and rotational motions: application to isotopomers of the hydrogen molecule

We have extended the multicomponent molecular orbital (MCMO) method to the full-configuration interaction (full-CI) fully variational molecular orbital method by elimination of translational and rotational motion components from total Hamiltonian. In the MCMO scheme, the quantum effects of protons and deuterons as well as electrons can be directly taken into account. All variational parameters in the full-CI scheme, i.e., exponents and centers (alpha and R) in the Gaussian-type function (GTF) basis set as well as the CI coefficients, are simultaneously optimized by using their analytical gradients. The total energy of the H(2) molecule calculated using the electronic [6s3p2d1f] and nuclear [1s1p1d1f] GTFs is -1.161 726 hartree, which can be compared to the energy of -1.164 025 hartree reported using a 512 term-explicitly correlated GTF calculation. Although the d- and f-type nuclear GTFs contribute to the improvement of energy convergence, the convergence of electron-nucleus correlation energy is slower than that of electron-electron one. The nuclear wave functions are delocalized due to the electron-nucleus correlation effect compared to the result of Hartree-Fock level of MCMO method. In addition, the average internuclear distances of all diatomic molecules are within 0.001 A of the previously reported experimental results. The dipole moment of the HD molecule estimated by our method is 8.4 x 10(-4) D, which is in excellent agreement with the experimental result of (8-10) x 10(-4) D.     

The multiscale coarse-graining method. IX. A general method for construction of three body coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for constructing a coarse-grained (CG) model of a system using data obtained from molecular dynamics simulations of the corresponding atomically detailed model. The formal statistical mechanical derivation of the method shows that the potential energy function extracted from an MS-CG calculation is a variational approximation for the true potential of mean force of the CG sites, one that becomes exact in the limit that a complete basis set is used in the variational calculation if enough data are obtained from the atomistic simulations. Most applications of the MS-CG method have employed a representation for the nonbonded part of the CG potential that is a sum of all possible pair interactions. This approach, despite being quite successful for some CG models, is inadequate for some others. Here we propose a systematic method for including three body terms as well as two body terms in the nonbonded part of the CG potential energy. The current method is more general than a previous version presented in a recent paper of this series [L. Larini, L. Lu, and G. A. Voth, J. Chem. Phys. 132, 164107 (2010)], in the sense that it does not make any restrictive choices for the functional form of the three body potential. We use hierarchical multiresolution functions that are similar to wavelets to develop very flexible basis function expansions with both two and three body basis functions. The variational problem is solved by a numerical technique that is capable of automatically selecting an appropriate subset of basis functions from a large initial set. We apply the method to two very different coarse-grained models: a solvent free model of a two component solution made of identical Lennard-Jones particles and a one site model of SPC/E water where a site is placed at the center of mass of each water molecule. These calculations show that the inclusion of three body terms in the nonbonded CG potential can lead to significant improvement in the accuracy of CG potentials and hence of CG simulations.