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.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

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     

A multireference configuration interaction method based on the separated electron pair wave functions

A multireference configurational interaction method based on the separated electron pair (SEP) wave functions, SEP‐CI approach, has been developed as an approximation to the traditional CASSCF method. It differs from the CASSCF method in that active orbitals are obtained from the SEP wave function without further optimization in the subsequent CI calculations, and the active space is automatically constructed according to the occupation coefficients of SEP natural orbitals. These features make the present SEP‐CI method computationally much less demanding than the CASSCF method. The applicability of the SEP‐CI method is illustrated with sample calculations on the insertion reaction of BeH₂ and dissociation energies of LiH, BH, FH, H₂O, and N₂. © 2005 Wiley Periodicals, Inc. J Comput Chem 27: 39–47, 2006     

Dependence of valency on geometry for configuration interaction wave functions

Earlier definitions of valencies of atoms, molecules, and molecular orbitals are extended to configuration interaction (CI ) wave functions. Using these definitions, valencies both at equilibrium and nonequilibrium geometries of molecules are calculated at the CI level and compared with non-CI results. CI valency correlation diagrams are obtained. Valency variation with bond length using correlated wave functions is found to behave properly unlike in the case of SCF wave functions.     

Cluster analysis of the full configuration interaction wave functions of cyclic polyene models

The first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI ) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.     

Extended Koopmans' theorem ionization potentials for beryllium atom shake-up transitions

The ionization potentials were calculated for Be using the extended Koopmans' theorem (EKT ) using several full configuration interaction (CI ) and multiconfigurational-self-consistent-field (MCSCF ) wave functions as reference wave functions. The wave functions used account for 89.7–96.7% of the correlation energy. Comparisons are made with experimental values and with δCI values calculated as the difference in energy obtained from CI wave functions for Be and Be⁺. The best EKT IP differed from the δCI value by 0.0003 eV for the lowest IP and by 0.0006 eV for ionization into the lowest ²P state of Be⁺. A calculation of ionization into the second ²P state of Be⁺ requires diffuse orbitals that are unimportant in the wave function for the ground state of Be. This results in small natural orbital occupation numbers for natural orbitals needed in the EKT calculation. © 1994 John Wiley & Sons, Inc.     

Nuclear-electronic orbital nonorthogonal configuration interaction approach

The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) approach is presented. In this framework, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear-electronic time-independent Schrodinger equation is solved with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wave function. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between the two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEO-full CI and Fourier grid calculations. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wave functions and the corresponding hydrogen tunneling splittings.     

Statistical electron correlation — coefficients and — holes in molecules

Summary Electron correlation in the H₂, LiH and BH molecules has been analyzed in terms of the statistical correlation coefficients introduced by Kutzelnigg, Del Re, and Berthier. Angular, radial (in-out), longitudinal (left-right) and transverse correlation coefficients have been evaluated from both self-consistent-field (SCF) and configuration interaction (CI) wave functions. It has been found that these coefficients reflect fairly well the correlation behavior in the molecular system. The lack of spherical symmetry in molecular densities adds new features to these correlation coefficients and this information can be useful for the study of electronic structure in molecules. The correlation hole function, Fermi and Coulomb holes in these systems have also been calculated and discussed.Dedicated to Professor Werner Kutzelnigg on the occasion of his sixtieth birthday     

A partially projected wave function for radicals

A partially projected wave function for odd electron systems with quantum number M=1/2, containing μu spin functions α and μ spin functions α, with fractional spin component αS_z=1/2 and 3/2 are derived from the totally projected wave function. To obtain these wave functions new symmetry relations between Sanibel coefficients for the odd electron case have been found, as well as the relations between primitive spin functions and their spin permutations. The wave function for the doublet state is shown not to contain contamination of the quadruplet state, and the wave function for the quadruplet does not have contamination of the duplet. Both wave functions exhibit equal forms except in the signs of their summation terms. The number of primitive spin functions depends on the number of electrons (n_s), it grows linearly as n_s=(N+3)/2. It can be considered as a generalization of the half projected Hartree–Fock wave function to the odd electron case. The HPHF wave function is defined for even electron systems and consists of only two Slater determinants, it has been shown to introduce some correlation effects and it has been successfully applied to calculate the low-lying excited states of molecules. Therefore, this investigation is the first step to propose a method to calculate the excited states of radicals when other methods are impracticable. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non relativistic continuum wave functions for two coulomb centres

We give the continuum wave function solutions to the Schrödinger equation for an electron moving field of two point nuclei, as an expansion in terms of one centre Coulomb wave functions in a prolate elliptical coordinate system. These solutions may be chosen to have a convenient asymptotic behaviour, and tend to the conventional solutions of the Helmholtz equation in the limit that the nuclear charge goes to zero. In symmetric systems, where both nuclei have the same charge the angular wave functions are found to be identical with those occurring in the free case, and the expansion coefficients for the corresponding radial solutions are given for selected values of electron energy and nuclear separation.[/p]     

General explanation of geometric phase effects in reactive systems: Unwinding the nuclear wave function using simple topology

We describe a simple topological approach which was used recently to explain geometric phase (GP) effects in the hydrogen-exchange reaction [Juanes-Marcos et al., Science 309, 1227 (2005)]. The approach is general and applies to any reactive system in which the nuclear wave function encircles a conical intersection (CI) and is confined to one adiabatic surface. The only numerical work required is to add and subtract nuclear wave functions computed with normal and GP boundary conditions. This is equivalent to unwinding the nuclear wave function onto a double cover space, which separates out two components whose relative sign is changed by the GP. By referring to earlier work on the Aharanov-Bohm effect, we show that these two components contain all the Feynman paths that follow, respectively, an even and an odd number of loops around the CI. These two classes of path are essentially decoupled in the Feynman sum, because they belong to different homotopy classes (meaning that they cannot be continuously deformed into one another). Care must be taken in classifying the two types of path when the system can enter the encirclement region from several different start points. This applies to bimolecular reactions with identical reagents and products, for which our approach allows a symmetry argument developed by Mead [J. Chem. Phys. 72, 3839 (1980)] to be generalized from nonencircling to encircling systems. The approach can be extended in order to unwind the wave function completely onto a higher cover space, thus separating contributions from individual winding numbers. The scattering boundary conditions are ultimately what allow the wave function to be unwound from the CI, and hence a bound state wave function cannot be unwound. The GP therefore has a much stronger effect on the latter than on the wave function of a reactive system.     

Méthode des Biorbitales: Application Préliminaire au Calcul de l'Energie de l'Etat Fondamental de l'Atome de Beryllium

The ground-state electronic energy of Be is calculated using the method of biorbitals (SCF –BI ). In this method the wave function is represented by an antisymmetrized product of identical pair functions. The basic set used to develop the biorbitals consists of the Watson s and p orbitals. The pair function is presumed to describe a singlet pair state. The energy associated with this function is minimized using a steepest descent procedure. A value of 0.0414 a.u. was found for the correlation energy, which is 44% of the total correlation energy. The SCF –BI method is compared with the CI method. The relationships are established between the expansion coefficients of both methods. The occupation numbers of orbitals are calculated.     

An application of perturbation theory ideas in configuration interaction calculations

Configuration interaction calculations of electronic wave functions for atoms and molecules have generally been limited to relatively small basis sets because of the exponential increase in the number of configurations as basis functions are added. While higher than quadruply excited configurations are of negligible importance in CI wave functions, it is shown that the effect of triple and quadruple excitation configurations can be substantially included even when the matrix elements between such configurations are neglected, leaving only their diagonal elements and the elements connecting them with the single and double excitations. This approximation is seen to be formally practically equivalent to a first-order perturbation expression for the wave function (second-order for the energy) based on an optimum linear combination of the zero, single, and double excitation configurations as the zero-order function. If suitable procedures are used, the amount of computational effort involved in such a calculation is roughly proportional to the fourth power of the number of basis functions employed, thus preventing the CI stage of the calculation from increasing in magnitude much faster than the stages involving the calculation and manipulation of the elementary integrals.     

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.     

A general nonlinear expansion form for electronic wave functions

A new expansion form is presented for electronic wave functions. The wave function is a linear combination of product basis functions, and each product basis function in turn is formally equivalent to a linear combination of configuration state functions that comprise an underlying linear expansion space. The expansion coefficients that define the basis functions are nonlinear functions of a smaller number of variables. The expansion form is appropriate for both ground and excited states and to both closed and open shell molecules. The method is formulated in terms of spin-eigenfunctions using the graphical unitary group approach (GUGA), and consequently it does not suffer from spin contamination.     

Molecular states of atoms. II

Orbital energy parameters, previously obtained from atomic valence state energies, are used in calculating approximate wave functions for their orbitals. The radial factors of these wave functions are expressed as linear combinations of three Gaussian type orbitals with selected exponents, the coefficients being determined by normalisation and reproduction of the kinetic energy and interelectron repulsion parameters. Wave functions of universal form are obtained for the non-transition elements up to xenon. Each calculated s orbital wave function (except 1s) has a radial node, as is appropriate if there is a p orbital in the same shell with none.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理(HVT)和Hellmann-Feynman 定理(HFT),导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出了由能级近似值计算定态波函数近似表达式的代数公式.给出了代数公式具体应用的几个典型一维势阱实例.此法可推广到二维势阱与三维势阱的情形.     

General algorithm for calculating vibrational–librational states of a rigid molecule in an external potential. Application to benzene–water complexes

A general algorithm of calculating the eigenstates of a rigid molecule trapped in an external potential is reported. The wave function and the potential are expanded about a common reference configuration. The expansion coefficients of the wave function are variationally determined. Contracted basis functions may be used to restrict the number of expansion coefficients. The use of the algorithm is illustrated by calculations of intermolecular eigenstates of benzene–water complexes.     

Ab initio MRD–CI calculations on cubane (neutral,carbocation, carboanion) and dissociation of nitrocubanes based on localized orbitals

Ab initio MRD –CI calculations based on localized orbitals were carried out for cubane (neutral, carbocation, carboanion) both in our customary MODPOT basis set and in an all-electron 4–31G basis set. The calculated MRD –CI charge distributions on C1 (the skeletal atom from which the H^? or H⁺ was removed) (ab initio MODPOT neutral 4.221, carbocation 3.796, carboanion 4.282; all-electron 4–31G neutral 6.171, carbocation 5.717, carboanion 6.078) indicate that the + or - charge does not remain localized on C1 but redistributes itself. This has significant implications for preparative reactions of energetically substituted cubanes. The MRD –CI population analyses differ somewhat from the SCF population analyses, especially in the calculated total overlap populations. To investigate this effect on electrostatic molecular potential contour (EMPC ) maps generated from SCF or MRD –CI wave functions, we wrote additional routines to calculate EMPC maps from MRD –CI wave functions. The EMPC maps generated from SCF or MRD –CI wave functions are different if the molecule needs an MRD –CI multideterminant wave function to describe it adequately. The EMPC map is a one-electron property. One-electron properties are derived from the 1-matrix. The 1-matrix is different for SCF or MRD –CI wave functions. Thus, all the one-electron properties (EMPC maps, population analyses, bond deviation indices, etc.) are different when calculated from SCF or MRD –CI wave functions if MRD –CI wave functions are necessary to describe a system properly. We calculate these one-electron properties from the 1-matrix from the final natural orbitals. Our preliminary calculations for the dissociation pathway indicate it takes more energy to dissociate a bond in 1-nitrocubane than in octanitrocubane. Even in their ground electronic states at equilibrium geometry, both 1-nitrocubane and octanitrocubane require MRD –CI wave functions to describe them properly. The c² of the single determinant SCF wave function is only 0.8401 for 1-nitrocubane and 0.8300 for octanitrocubane. There are contributions from skeletal excitations as there are for cubane itself as well as excitations involving the nitrogroup. As the bond in nitrocubane is dissociated to 8.00 bohrs, the c² of the SCF contribution drops to only 0.4606 (1-nitrocubane) and 0.4445 (octanitrocubane). At this C₁? N₁ intermolecular distance, the largest excitations are in the C₁? N₁ bond: (C₁? N₁)² → (C₁? N₁*)², (C₁? N₁) → (C₁? N₁*). We also calculated the first electronically excited state for the dissociation pathway for selected points for both 1-nitrocubane and octanitrocubane.