Geometrical properties of nodal surfaces of many‐electron wave functions

Hypothesis of the exclusion of equipotential surfaces for many‐electron wave functions (MWF) has been enunciated. This hypothesis clarifies the physical meaning of the Pauli exclusion principle and opens the way for future progress of new quantum‐chemical methods for the construction of approximate MWFs differing from the traditional Hartree–Fock approximation. The equipotential surface exclusion principle has been tested on traditional representative “test systems” of quantum mechanics: the helium atom, the lithium atom, and the hydrogen molecule. Judging by the results of these tests, the use of the suggested approach can lead to a considerable increase in the efficiency of high‐accuracy quantum‐chemical calculations. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Geometrical linear responses and directional energy derivatives for energetically degenerate MCSCF electronic functions

Summary For state-averaged multiconfigurational self consistent field (SA-MCSCF) wave functions, second-order geometrical response equations are derived that allow the determination of first-order configuration amplitude response for equally weighted, energetically degenerate states. The first-order response equations obtained in earlier work do not suffice to determine these particular responses parameters. To formulate such a derivation in a well defined manner, it is found that a specific linear combination of the degenerate states must be formed; this specific combination of states then defines how state energies and wave functions evolve as one passes through the surface intersection. The linear combination among the degenerate states is dependent upon the molecular distortion for which the responses are to be evaluated. Expressions for first- and second-order directional energy derivatives for these energetically degenerate wave functions are also derived. All the equations obtained are computationally tractable and expressed in terms of quantities that result from optimizing the SA-MCSCF wave functions and from solving the first- and part of the second-order geometrical response equations.     

Generalization of analytic energy derivatives for configuration interaction wave functions

This paper takes the form of a review including some original contributions. A fresh derivation of analytic energy derivative expressions for configuration interaction (CI) wave functions is presented. In this method the CI energy is described by _IJC_IC_J(H _IJ-_IJE) so that the orthonormality condition is explicitly included therein. In the sequence of differentiations up to fourth order it will be demonstrated that each derivative may be expressed in terms of (H _IJ-_IJE) and its derivatives in a symmetric way with respect to the interchange of differential variables. In a similar manner, the CI variational condition may be described in an equation which explicitly includes the normalization condition. It is shown that the differentiation of the modified variational condition produces the coupled perturbed configuration interaction (CPCI) equations in directly soluble and compact forms. The necessary formulae for the energy derivatives up to fourth order and the CPCI equations up to second order are explicitly given.     

Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.     

Correlation energy extrapolation by intrinsic scaling. III. Compact wave functions

The information gained in the context of extrapolating the correlation energy by intrinsic scaling is used to shorten the full configurational expansions of electronic wave function without compromising their chemical accuracy. The truncations are accomplished by judiciously limiting the participation of the ranges of predetermined approximate sets of natural orbitals in the various excitation categories.     

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.     

Efficient calculation of potential energy surfaces for the generation of vibrational wave functions

An automatic procedure for the generation of potential energy surfaces based on high level ab initio calculations is described. It allows us to determine the vibrational wave functions for molecules of up to ten atoms. Speedups in computer time of about four orders of magnitude in comparison to standard implementations were achieved. Effects due to introduced approximations--within the computation of the potential--on fundamental modes obtained from vibrational self-consistent field and vibrational configuration interaction calculations are discussed. Benchmark calculations are provided for formaldehyde and 1,2,5-oxadiazole (furazan).     

Excited state wave functions

Wave functions and energies were calculated for the 2s, 3p ₀, and 4d ₀ states of the hydrogen atom using the Messmer and Rayleigh-Ritz variational methods with minimization of the second eigenvalue. The wave functions were linear expansions of Gaussian functions and both linear and exponential parameters were varied. Except for the two term expansions, calculated values of the energies and expectation values, r ^–1, r and r ² were within two percent of the true values for both methods.     

Optimal Hylleraas wave functions

Hylleraas wave functions composed of the optimally combinedN terms (2 N 20) are presented for two-electron atoms with nuclear chargesZ = 1 (H^–), 2(He), 3(Li⁺), 5(B³⁺), and 10(Ne⁸⁺). The spherically-averaged electron density (r) and electron-pair densityh(r ₁₂) are constructed in a simple and analytical functional form from the 20-term functions. Comparison of several one- and two-electron moments r ^k and r ₁₂ ^k shows that the present density functions have near-exact accuracy.     

Two by two rotations of orbitals for minimizing the energy of LCAO-MO wave functions

A new method is studied for finding the molecular orbitals which minimize the energy of an LCAO-MO wavefunction. The method makes use of successive rotations of pairs of orbitals. It can be applied to multi-determinant as well as to single-determinant wavefunctions. Criteria are found to minimize excited states.

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Zusammenfassung Es wird eine neue Methode zur Bestimmung der M.O.s nach dem Kriterium minimaler Gesamtenergie beschrieben. Dabei werden jeweils zwei M.O.s sukzessive transformiert. Das Verfahren bleibt auch dann anwendbar, wenn für die Zustandsfunktion eine Linearkombination von HS-Determinanten angesetzt wird. Zum Schluß werden noch einige Kriterien für die Bestimmung von angeregten Zuständen angegeben.

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Résumé Une nouvelle méthode est étudiée ici pour trouver les orbitales moléculaires qui minimisent l'énergie d'une fonction d'onde LCAO-MO. La méthode utilise des rotations successives de couples d'orbitales. La méthode peut minimiser aussi des fonctions d'onde multi-déter-minantales et les états excités.

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The author is indebted with Professor B. Pullman and Dr. A. Pullman for their suggestions and helpful advises. He aknowledges also Dr. Berthier for useful discussions.

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This work prepared in part at the Institut de Biologie Physico-Chimique, Université de Paris, was supported by a grant CY-3073 of the US Public Health Service (National Cancer Institute) to that Institute.     

Constrained self-consistent-field wave functions with improved long-range behavior

The recent suggestion that the long-range behavior of energy-optimized Gaussian basis sets can be improved by augmenting them with a Gaussian chosen to satisfy a constraint involving a linearly averaged position moment is explored. Calculations indicate that the high-order moments 〈r^k〉, with k > 4, in He, Be, and Li^?, and 〈x^kz^L?k〉, with L > 4 and k ≤ L, in H₂ are improved by the constraint, but that lower-order moments and dipole polarizabilities are not. In H₂, the higher moments with a given L improve by different amounts for different k, and, hence, the multipole moments do not improve. The basis-set superposition error in He? He and Be? Be interaction energy calculations decreases if the internuclear distance is large enough. Thus, the constraint procedure improves the very long range behavior of the self-consistent-field wave functions. © 1992 John Wiley & Sons, Inc.     

Laplace transform wave functions

The wave function defining a quantum-mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen-like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.     

Analytic evaluation of infrared intensities and polarizabilities by two-configuration self-consistent field wave functions

For more than a few molecular electronic states, the simplest qualitatively correct picture of the electronic structure is provided by the two-configuration self-consistent-field (TCSCF) method. Here, analytic methods are reported for the evaluation of TCSCF infrared intensities and polarizabilities. These new methods have been implemented and applied to the molecules CH₂(¹A₁), CF₂, CH ₃ ^– , NH₃, HF and O₃. Nine different basis sets, ranging from 3–21G to triple zeta plus double polarization (TZ + 2P), have been used. In several cases one finds qualitative differences between the analogous SCF and TCSCF predictions.     

CN~+分子势能函数和光谱常数的多参考组态相互作用方法研究

采用多参考组态相互作用方法和aug-cc-p V5Z基函数组计算了CN+分子11∑+,21∑+,13∑+和13Π电子态的势能曲线。利用MS势能函数拟合得到了相应的解析势能表达式。在此基础上求解CN+分子的核运动薛定谔方程,获得了全部振动和转动能级,并用Dunham系数展开式拟合出了光谱常数,与目前仅有的11∑+,21∑+态的文献报道结果进行了比较。结果可对航天尾气及工业过程光谱方法监控提供参考。     

MRCI potential energy curves and spectroscopic constants of the ground and low-lying excited states of HgCd

The multireference configuration interaction (MRCI) electronic energy calculations with different basis sets have been performed on the ground state (X¹Σ) and three low-lying excited states (³Σ, ¹Π and ³Π) of HgCd dimer. The obtained potential energy curves (PECs) are fit to analytical potential energy functions (APEFs) using the Murrell–Sorbie potential function. Spectroscopic constants are calculated using the APEFs. Based on the PECs, the vibrational levels of each state are predicted. Our equilibrium positions of the X¹Σ state and ³Π state are in excellent agreement with the experimental reports.     

Systematic errors in the total energy of molecular wave functions calculated within the PRDDO approximations

We compare calculated total energies for 150 open-chain molecules using ab initio methodology and the PRDDO approximations. The bulk of the errors implicit in the PRDDO approximations are apparently of a one-center nature, i.e., they are due to the number and type of atoms in the molecule, and not the details of the molecular geometry. Atomic correction factors are developed which reduce the errors in the calculated total energy of PRDDO wave functions by a factor of eight relative to the ab initio reference calculations. PRDDO calculations on ring and cage compounds are shown to have additional systematic errors in the total energy.     

Analytic expression of the second derivatives of electronic energy for full configuration interaction wave functions

Summary A new analytic second derivative expression of the electronic energy is derived for full configuration interaction (CI) wave functions. This formula is shown to be free from the derivative terms of both CI and MO coefficients. The second-order relationships between CI and MO coefficients for full CI wave functions are also presented.