Electron pair localization function: a practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called "electron pair localization function," is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.     

Visualizing molecular wavefunctions using Monte Carlo methods

Using explicitly correlated wavefunctions and variational Monte Carlo we calculate the electron density, the electron density difference, the intracule density, the extracule density, two forms of the kinetic energy density, the Laplacian of the electron density, the Laplacian of the intracule density, and the Laplacian of the extracule density on a dense grid of points for the ground state of the hydrogen molecule at three internuclear distances (0.6, 1.4, 8.0). With these values we construct a contour plot of each function and describe how it can be used to visualize the distribution of electrons in this molecule. We also examine the influence of electron correlation on each expectation value by calculating each function with a Hartree–Fock wavefunction and then comparing these values with our explicitly correlated values. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

A simultaneous probability density for the intracule and extracule coordinates

We introduce the intex density X(R,u), which combines both the intracular and extracular coordinates to yield a simultaneous probability density for the position of the center-of-mass radius (R) and relative separation (u) of electron pairs. One of the principle applications of the intex density is to investigate the origin of the recently observed secondary Coulomb hole. The Hartree-Fock (HF) intex densities for the helium atom and heliumlike ions are symmetric functions that may be used to prove the isomorphism 2I(2R)=E(R), where I(u) is the intracule density and E(R) is the extracule density. This is not true of the densities that we have constructed from explicitly correlated wave functions. The difference between these asymmetric functions and their symmetric HF counterparts produces a topologically rich intex correlation hole. From the intex hole distributions (X(exact)(R,u)-X(HF)(R,u)), we conclude that the probability of observing an electron pair with a very large interelectronic separation increases with the inclusion of correlation only when their center-of-mass radius is close to half of their separation.     

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.     

Electron affinities with diffusion quantum Monte Carlo for C2 and BO molecules

To verify the performance of the fixed-node diffusion quantum Monte Carlo method in electron affinities calculations, the adiabatic electron affinities of C(2) and BO molecules calculated by the fixed-node Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals are presented in this work. The random walk based results, 3.264(43) and 2.507(32) eV for C(2) and BO, respectively, are compared with the available best experimental determined values, 3.269(6) and 2.508(8) eV for C(2) and BO, respectively, and the results of other theoretical calculations.     

Zero-variance zero-bias quantum Monte Carlo estimators for the electron density at a nucleus

We derive new quantum Monte Carlo (QMC) estimators for the electronic density at the position of a point nucleus using the zero-variance and zero-bias principles. The resulting estimators are highly efficient, and are significantly simpler to implement and use than alternative methods, as they contain no adjustable parameters. In addition, they can be used in both variational and diffusion QMC calculations. Our best estimator is used to calculate the most accurate available estimates of the total electron density at the nucleus for the first-row atoms Li-Ne, the Ar atom, and the diatomic molecules B(2), N(2), and F(2).     

Diffusion quantum Monte Carlo for equilibrium structures and harmonic frequencies of ethane and ozone molecules

Application of the Ornstein-Uhlenbeck diffusion quantum Monte Carlo method in combination with a trial wave function constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals to studies on the equilibrium structures and harmonic frequencies of ethane and ozone is presented. These Monte Carlo computed results are compared with those of experiments as well as the coupled cluster methods with the correlation consistent basis sets for the two molecules. For ozone, we also compare the Monte Carlo results with the results from multireference calculations.     

Energies of the first row atoms from quantum Monte Carlo

All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.     

Direct optimization of nodal hypersurfaces in approximate wave functions

The fixed-node variant of the diffusion quantum Monte Carlo method (FN-DMC) is capable of obtaining the exact eigenvalues (albeit numerically with statistical error) of a many-electron Hamilton operator, provided that the nodal hypersurface of the exact wave function is given. The use of nodes of a trial wave function leads to the node location error. The authors have developed local criteria to assess the accuracy of the nodes based on the distances of the nodal hypersurfaces of PsiT, TPsiT, and HPsiT which coincide for the exact wave function. These criteria are used to develop direct optimization methods for the nodal hypersurface. The optimization of the nodes is demonstrated for simple wave functions of the Be atom and the C2 molecule and verified with FN-DMC calculations.     

Quantum Monte Carlo calculations of the potential energy curve of the helium dimer

We report results of two quantum Monte Carlo methods -- variational Monte Carlo and diffusion Monte Carlo -- on the potential energy curve of the helium dimer. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all.     

量子Monte Carlo固定节面法试探函数研究

量子ＭｏｎｔｅＣａｒｌｏ固定节面法试探函数研究黄宏新曹泽星（湖南师范大学化学系长沙４１００８１）关键词量子ＭｏｎｔｅＣａｒｌｏ方法试探函数奇点条件固定节面量子ＭｏｎｔｅＣａｒｌｏ（ＦＮＱＭＣ）方法是ＱＭＣ方法中使用最多的一种，它的试探函数一般是这样组...     

Optimum and efficient sampling for variational quantum Monte Carlo

Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.     

Path-integral virial estimator for reaction-rate calculation based on the quantum instanton approximation

The quantum instanton approximation is a type of quantum transition-state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low-order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat-capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial (-type) estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.     

Rydberg states with quantum Monte Carlo

Calculations on Rydberg states are performed using quantum Monte Carlo methods. Excitation energies and singlet-triplet splittings are calculated for two model systems, the carbon atom (3P and 1P) and carbon monoxide ((1Sigma and 3Sigma). Kohn-Sham wave functions constructed from open-shell localized Hartree-Fock orbitals are used as trial and guide functions. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method depends strongly on the wave function's nodal hypersurface. Nodal artefacts are investigated for the ground state of the carbon atom. Their effect on the FN-DMC results can be analyzed quantitatively. FN-DMC leads to accurate excitation energies but to less accurate singlet-triplet splittings. Variational Monte Carlo calculations are able to reproduce the experimental results for both the excitation energies and the singlet-triplet splittings.     

Bond breaking with auxiliary-field quantum Monte Carlo

Bond stretching mimics different levels of electron correlation and provides a challenging test bed for approximate many-body computational methods. Using the recently developed phaseless auxiliary-field quantum Monte Carlo (AF QMC) method, we examine bond stretching in the well-studied molecules BH and N(2) and in the H(50) chain. To control the sign/phase problem, the phaseless AF QMC method constrains the paths in the auxiliary-field path integrals with an approximate phase condition that depends on a trial wave function. With single Slater determinants from unrestricted Hartree-Fock as trial wave function, the phaseless AF QMC method generally gives better overall accuracy and a more uniform behavior than the coupled cluster CCSD(T) method in mapping the potential-energy curve. In both BH and N(2), we also study the use of multiple-determinant trial wave functions from multiconfiguration self-consistent-field calculations. The increase in computational cost versus the gain in statistical and systematic accuracy are examined. With such trial wave functions, excellent results are obtained across the entire region between equilibrium and the dissociation limit.     

One and two body densities for excited states of the helium confined atom

A study of the first excited states of the helium atom confined under impenetrable spherical walls is carried out. Both single particle and two body, intracule and extracule, densities are constructed. Crossing levels and Hund's rule are analyzed in terms of the contribution to the total energy from kinetic, electron–nucleus, and electron–electron energies. A study about the behavior of the single particle and two body densities is carried out. The Multiconfiguration Parameterized Optimized Effective Potential method is employed with a cut-off factor to account for Dirichlet boundary conditions. Single particle density is analytically constructed whereas the Monte Carlo algorithm is used to calculate two body densities.     

The lithium-thiophene riddle revisited

A recent study of the interaction of a lithium atom with the thiophene molecule found a large disagreement between high-level coupled cluster (CCSD(T)/AVTZ) and quantum Monte Carlo (fixed-node diffusion Monte Carlo, or FNDMC) calculations. We address this "lithium-thiophene riddle" by analyzing the influence of crucial FNDMC simulation parameters, namely, the one-electron models, basis sets, and pseudopotentials used for the generation of the trial wave function. These are shown to have a significant impact on the calculated FNDMC interaction energies, and good agreement between CCSD(T) and FNDMC is found when nodal hypersurfaces of sufficient quality are used. On the basis of our proposed consensus reference value, we go on to benchmark the standard toolbox of lower-level quantum chemistry methods for this model interaction. Newly developed dispersion-corrected DFT methods perform reasonably well despite the partial charge transfer character of the interaction and might well be worthy of further study in larger lithium-thiophene systems.     

Including quantum subsystem character within classical equilibrium simulations

A mixed quantum/classical density matrix approximation is derived. The density matrix makes use of quantum subsystem vibrational wave functions. The diagonal of the density matrix can be used as an equilibrium distribution in Monte Carlo simulations. The approximate distribution compares well with the path integral distribution for a model system. Since it includes quantum subsystem information, it performs much better than the quadratic Feynman-Hibbs distribution. These types of distributions can aid in including quantum vibrational information in otherwise classical simulations.     

Quantum Monte Carlo study of the Ne atom and the Ne+ ion

We report all-electron and pseudopotential calculations of the ground-state energies of the neutral Ne atom and the Ne(+) ion using the variational and diffusion quantum Monte Carlo (DMC) methods. We investigate different levels of Slater-Jastrow trial wave function: (i) using Hartree-Fock orbitals, (ii) using orbitals optimized within a Monte Carlo procedure in the presence of a Jastrow factor, and (iii) including backflow correlations in the wave function. Small reductions in the total energy are obtained by optimizing the orbitals, while more significant reductions are obtained by incorporating backflow correlations. We study the finite-time-step and fixed-node biases in the DMC energy and show that there is a strong tendency for these errors to cancel when the first ionization potential (IP) is calculated. DMC gives highly accurate values for the IP of Ne at all the levels of trial wave function that we have considered.