Wave Function Optimization Research Articles

The effect of discontinuous wave functions on optimizations of VMC calculations of quantum systems

Here, a relatively good comparison has been done between the performance results of discontinuous and continuous forms of wave function on quantum systems as well. In this project, we’ve estimated the energy of ground and first state as the expectation value of Hamiltonian of harmonic oscillator and binding energy of nucleons confined in a Yukawa potential by making use of following three wave function: exponential, linear and Gaussian form which are convenient to optimize and may be evaluated rapidly are devised and tested and we stress the advantages of using expressions which are Gaussian in the variable parameters. Exponential function is quite different from a Gaussian ones as it has a cusp at x=0, hence a discontinuity in its derivatives. We then have performed to test inaccuracy of the discontinuous wave function and energy’s error of its exact value because using of exponential and linear function by mathematical statements of calculation of delta and Heaviside function and either VMC calculation of wave function optimization, as both of them will have the same result of error value. The result of our study is very promising for future applications of quantum Monte Carlo methods especially in many body systems

Collisional shift of the heavy atoms hyperfine lines in an atmosphere of the inert gas

New relativistic approach, based on the relativistic many-body perturbation theory (PT) with using the optimized wave functions sets, is applied to calculating the potentials and hyper fine structure collision shift for heavy atoms in atmosphere of inert gases. Data for the collisional shifts of the Cs, Tl and Yb atoms in atmosphere of the inert gases are presented.

An approach to evaluation of 19F‐NMR chemical shifts via basis functions analysis in fluorinated small compounds

AbstractAn extensive GIAO (gauge‐including atomic orbital) calculation has been made within Hartree‐Fock (HF), density functional theory (DFT) and second‐order Møller−Plesset perturbation theory (MP2), in conjunction with selected basis sets for the prediction of 19F chemical shift values of 26 different F nuclei in small fluorine‐containing molecules. The effect of four factors, namely, electron correlation treatment, triple‐ξ valence shell, diffuse function and polarization function were assessed using a systematic comparison of the results. Based on types of series, the optimized wave functions were proposed for different molecules. An additional comparison of the principal components of the shift tensor has been used as a supplementary study to confirm the validity of the results. For the series of molecules studied, the diffuse function is the most effective factor used in this applied model and the presence of polarization function is important to minimize the difference between theoretical and experimental values. By including the electron correlation treatment, DFT approach with large basis sets provides the most reliable results. In addition, two 24 factorial designs were considered to assess the reliability of applied scheme for selection the most efficient basis function and the results were discussed in this procedure. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40A: 192–204, 2012.

Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory

Range-separated density-functional theory combines wave function theory for the long-range part of the two-electron interaction with density-functional theory for the short-range part. When describing the long-range interaction with non-variational methods, such as perturbation or coupled-cluster theories, self-consistency effects are introduced in the density functional part, which for an exact solution requires iterations. They are generally assumed to be small but no detailed study has been performed so far. Here, the authors analyze self-consistency when using Møller-Plesset-type (MP) perturbation theory for the long range interaction. The lowest-order self-consistency corrections to the wave function and the energy, that enter the perturbation expansions at the second and fourth order, respectively, are both expressed in terms of the one-electron reduced density matrix. The computational implementation of the latter is based on a Neumann series which, interestingly, even though the effect is small, usually diverges. A convergence technique, which perhaps can be applied in other uses of Neumann series in perturbation theory, is proposed. The numerical results thus obtained show that, in weakly bound systems, self-consistency can be neglected since the long-range correlation does not affect the density significantly. Although MP is not adequate for multireference systems, it can still be used as a reliable analysis tool. Though the density change is not negligible anymore in such cases, self-consistency effects are found to be much smaller than long-range correlation effects (less than 10% for the systems considered). For that reason, a sensible approximation might be to update the short-range energy functional term while freezing its functional derivative, namely, the short-range local potential, in the wave function optimization. The accuracy of such an approximation still needs to be assessed.

Rayleigh–ritz method for excited quantum states via nonlinear variations without constraints: Role of supersymmetry

AbstractQuantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited‐state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited‐state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

The prediction of amino proton chemical shifts using optimally selected wave function

AbstractGas phase amino proton chemical shifts in the 54 of amines have been predicted using Gauge‐independent atomic orbital (GIAO) method and optimally selected wave function. The effects of electron correlation, triple‐ξ valance shell, diffuse function, and polarization function on calculated amino proton chemical shifts have been investigated using factorial design as a multivariate technique. Different optimized wave functions for different groups of amines were recommended. A wave function as the best level of the theory is proposed for homologue amines covered. In this context, B3LYP/6‐311+G and HF/6‐311+G wave functions have been recommended as the best and the most efficient level of theory for calculating amino proton chemical shifts of alkylamines and arylamines respectively. © 2011 Wiley Periodicals, Inc. Concepts Magn Reson Part A 38A: 25–32, 2011.

Strong correlation treated via effective hamiltonians and perturbation theory

We propose a new approach to determine a suitable zeroth-order wavefunction for multiconfigurational perturbation theory. The same ansatz as in complete active space (CAS) wavefunction optimization is used but it is split in two parts, a principal space (A) and a much larger extended space (B). Löwdin's partitioning technique is employed to map the initial eigenvalue problem to a dimensionality equal to that of (A) only. Combined with a simplified expression for the (B) portion of the wavefunction, we are able to drastically reduce the storage and computational demands of the wavefunction optimization. This scheme is used to produce reference wavefunctions and energies for subsequent second-order perturbation theory (PT2) corrections. Releasing the constraint of computing the exact CAS energy and wavefunction prior to the PT2 treatment introduces a nonstandard paradigm for multiconfigurational methods. Based on the results of test calculations, we argue that principal parts with only few percents of the total number of CAS configurations could provide final multiconfigurational PT2 energies of the same accuracy as in the standard paradigm. In the future, algorithmic improvements for this scheme will bring into reach active spaces much beyond the present limit of CAS-based methods, therefore allowing for accurate studies of systems featuring strong correlation.

An efficient recursive algorithm to compute wave function optimization gradients for the graphically contracted function method

AbstractAn efficient recursive algorithm is presented to compute orbital‐level Hamiltonian matrices for wave functions expanded in a basis of graphically contracted functions (GCF). Each GCF depends on a nonlinear set of parameters called arc factors. The orbital‐level Hamiltonian matrices characterize the dependence of the energy on the wave function changes associated with a subset of these nonlinear parameters corresponding to an individual molecular orbital. From these Hamiltonian matrices, gradients with respect to the nonlinear arc factor parameters may be computed and other arc factor optimization algorithms may be used. The recursive algorithm allows the orbital‐level Hamiltonian matrices to be computed with O(Nωn4) total effort, where NGCF is the dimension of the GCF basis, n is the dimension of the orbital basis, and where the scale factor ω depends on the number of electrons N and ranges from O(N0) to O(N2) depending on the complexity of the underlying Shavitt graph. This effort is about two to five times that required to compute an energy expectation value for a given set of arc factors; thus the energy and gradient have the same scaling behavior with increasing molecule size, NGCF dimension, and orbital basis size. Timings are given for wave functions that correspond to configuration state function expansions over 1073 in length, many orders of magnitude larger than can be considered using traditional electronic structure methods. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Aspects of Entanglement in Quantum Many-Body Systems

Knowledge of the entanglement properties of the wave functions commonly used to describe quantum many-particle systems can enhance our understanding of their correlation structure and provide new insights into quantum phase transitions that are observed experimentally or predicted theoretically. To illustrate this theme, we first examine the bipartite entanglement contained in the wave functions generated by microscopic many-body theory for the transverse Ising model, a system of Pauli spins on a lattice that exhibits an order-disorder magnetic quantum phase transition under variation of the coupling parameter. Results for the single-site entanglement and measures of two-site bipartite entanglement are obtained for optimal wave functions of Jastrow-Hartree type. Second, we address the nature of bipartite and tripartite entanglement of spins in the ground state of the noninteracting Fermi gas, through analysis of its two- and three-fermion reduced density matrices. The presence of genuine tripartite entanglement is established and characterized by implementation of suitable entanglement witnesses and stabilizer operators. We close with a broader discussion of the relationships between the entanglement properties of strongly interacting systems of identical quantum particles and the dynamical and statistical correlations entering their wave functions.

An Active-Site Phenylalanine Directs Substrate Binding and C−H Cleavage in the α-Ketoglutarate-Dependent Dioxygenase TauD

Enzymes that cleave C-H bonds are often found to depend on well-packed hydrophobic cores that influence the distance between the hydrogen donor and acceptor. Residue F159 in taurine alpha-ketoglutarate dioxygenase (TauD) is demonstrated to play an important role in the binding and orientation of its substrate, which undergoes a hydrogen atom transfer to the active site Fe(IV)=O. Mutation of F159 to smaller hydrophobic side chains (L, V, A) leads to substantially reduced rates for substrate binding and for C-H bond cleavage, as well as increased contribution of the chemical step to k(cat) under steady-state turnover conditions. The greater sensitivity of these substrate-dependent processes to mutation at position 159 than observed for the oxygen activation process supports a previous conclusion of modularity of function within the active site of TauD (McCusker, K. P.; Klinman, J. P. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 19791-19795). Extraction of intrinsic deuterium kinetic isotope effects (KIEs) using single turnover transients shows 2- to 4-fold increase in the size of the KIE for F159V in relation to wild-type and F159L. It appears that there is a break in behavior following removal of a single methylene from the side chain of F159L to generate F159V, whereby the protein active site loses its ability to restore the internuclear distance between substrate and Fe(IV)=O that supports optimal hydrogenic wave function overlap.

Variational geminal-augmented multireference self-consistent field theory: Two-electron systems

We introduce a geminal-augmented multiconfigurational self-consistent field method for describing electron correlation effects. The approach is based on variational optimization of a MCSCF-type wave function augmented by a single geminal. This wave function is able to account for some dynamic correlation without explicit excitations to virtual molecular orbitals. Test calculations on two-electron systems demonstrate the ability of the proposed method to describe ionic and covalent electronic states in a balanced way, i.e., including the effects of both static and dynamic correlation simultaneously. Extension of the theory to larger systems will potentially provide an alternative to standard multireference methods.

Calculation of geometrically active atomic state

We develop a computer code to calculate ab-initio variational configuration interaction of electronic structure of atoms via generalised Lagurre type orbitals. Treating the orbital effective charges as variational parameters and computing the absolute minimum of energy, yield its optimal wave function.Then utilizing the one-electron density and the probability distribution of the angular two-electron density according to the optimal wave function, we investigate the geometrically active atomic states (GAASs) of Be, B, C, N, O and Ne atoms that are in the first excited states with configurations sαpβ. These results reveal that as an intrinsic property of the wave function of atoms, the angle of the most probabl angular distribution of two-electron density is approximately equal to the bond angle of the molecule, which usually can be explained by the hybridization theory.

Optimization of Many-Body Wave Function

Optimization of Many-Body Wave Function

Obtaining Hartree–Fock and density functional theory doubly excited states with Car–Parrinello density matrix search

The calculation of doubly excited states is one of the major problems plaguing the modern day excited state workhorse methodology of linear response time dependent Hartree-Fock (TDHF) and density function theory (TDDFT). We have previously shown that the use of a resonantly tuned field within real-time TDHF and TDDFT is able to simultaneously excite both the alpha and beta electrons to achieve the two-electron excited states of minimal basis H(2) and HeH(+) [C. M. Isborn and X. Li, J. Chem. Phys. 129, 204107 (2008)]. We now extend this method to many electron systems with the use of our Car-Parrinello density matrix search (CP-DMS) with a first-principles fictitious mass method for wave function optimization [X. Li, C. L. Moss, W. Liang, and Y. Feng, J. Chem. Phys. 130, 234115 (2009)]. Real-time TDHF/TDDFT is used during the application of the laser field perturbation, driving the electron density toward the doubly excited state. The CP-DMS method then converges the density to the nearest stationary state. We present these stationary state doubly excited state energies and properties at the HF and DFT levels for H(2), HeH(+), lithium hydride, ethylene, and butadiene.

Gauge origin independent calculations of nuclear magnetic shieldings in relativistic four-component theory

The use of perturbation-dependent London atomic orbitals, also called gauge including atomic orbitals, has proven efficient for calculations of NMR shielding constants and other magnetic properties in the nonrelativistic framework. In this paper, the theory of London atomic orbitals for NMR shieldings is extended to the four-component relativistic framework and our implementation is described. The relevance of London atomic orbitals in four-component calculations as well as computational aspects are illustrated with test calculations on hydrogen iodide. We find that the use of London atomic orbitals is an efficient method for reliable calculations of NMR shielding constants with standard basis sets, also for four-component calculations with spin-orbit coupling effects included in the wave function optimization. Furthermore, we find that it is important that the small component basis functions fulfill the magnetic balance for accurate description of the diamagnetic shielding and that the role of London atomic orbitals in the relativistic domain is to provide atomic magnetic balance even in the molecular case, thus greatly improving basis set convergence. The Sternheim approximation, which calculates the diamagnetic contribution as an expectation value, leads to significant errors and is not recommended.

Car–Parrinello density matrix search with a first principles fictitious electron mass method for electronic wave function optimization

In spite of its success in molecular dynamics and the advantage of being a first order propagation technique, the Car-Parrinello method and its variations have not been successful in self-consistent-field (SCF) wave function optimization due to slow convergence. In this article, we introduce a first principles fictitious mass scheme to weigh each individual density element differently and instantaneously. As an alternative to diagonalization in SCF, the Car-Parrinello scheme is implemented as a density matrix search method. Not only does the fictitious mass scheme developed herein allow a very fast SCF convergence, but also the Car-Parrinello density matrix search (CP-DMS) exhibits linear scaling with respect to the system size for alanine helical chain test molecules. The excellent performance of CP-DMS holds even for very challenging compact three-dimensional quantum particles. While the conventional diagonalization based SCF method has difficulties optimizing electronic wave functions for CdSe quantum dots, CP-DMS shows both smooth and faster convergence.

Quadratically convergent simultaneous optimization of wavefunction and geometry

A second-order (quadratically convergent) method is presented for simultaneously optimizing the molecular wavefunction and geometry within the finite basis Hartree–Fock model. Due to inclusion of exact second derivative information, its convergence properties are greatly superior to first-order optimization schemes, while its cost only scales as the fourth power of the molecular size, like conventional first-order optimizations. By contrast, a conventional second-order geometry optimization (in which the wavefunction is constrained to be optimal), involves a step scaling as the fifth power. The performance of the new method relative to a conventional geometry optimization is illustrated by calculations on (S)-α-formylaminopropanamide, CHO—NH—CHCH3—CO—NH2.

Multiconfigurational wavefunction optimization using the unitary group method

We have successfully used graphical unitary group methods to implement efficiently both multiconfigurational self-consistent field and configuration interaction ab initio quantum chemical computer programs. We indicate how these group theoretic methods improve efficiency and we demonstrate their utility on the ground and excited states of a model problem for which a full-configuration interaction calculation can be performed.

Range relaxation 2. Determination of a reaction coordinate over an energy surface of several dimensions

This paper introduces the pass function, the second of three “relaxation” functions to be used in a general functional formulation of parameters necessary for the a priori calculation of molecular structure and dynamics. The approach is intended to simplify the calculation of a reaction coordinate in a complex many-dimensional energy surface. In principle it involves the minimisation of a path integral across a transformed energy surface; and it converges on an analytical expression for the pass of minimum energy across the surface, and the optimal wavefunction of any given form at each point along that pass. In practice numerical evaluation of the integral by Gauss-Legendre quadrature necessitates calculation of the energy at a small number of nuclear configurations. A bridge may be formed with experiment by allowing possible use of empirical expressions or limiting molecular data as a guide to make the optimisation more rapid, though the method converges on a non-empirical result. The scheme is applied here to the hydrogen exchange reaction. For such a two dimensional surface the labour involved is estimated as roughly 5%-15% of that required for a conventional calculation of comparable scope. The economy increases with the number of dimensions considered. The approach can be applied in conjunction with the use of any type of molecular wavefunction.

Optimized Wannier Functions for Hubbard Chain

One-dimensional atomic chain with a variable-range hopping is described within the extended Hubbard model. The Gutzwiller-ansatz approximation is used to determine the optimized single-particle (Wannier) wave functions in the correlated state. Hopping integral up to the third neighbors is taken into account and the results are compared with those for the inflnite hopping range. Ground state energy of the system is compared with that making use of the rigorous Lieb{Wu solution with the optimized wave functions. The evolution of the properties as a function of interatomic distance is discussed.