MCSCF Wave Functions Research Articles

Quantum fluid dynamics of many‐electron systems in three‐dimensional space

AbstractIn presence of external electric and magnetic fields, the Schrödinger equation for many‐electron systems is transformed into a continuity equation and an Euler‐type equation of motion in configuration space. Then, using the natural‐orbital Hamiltonian, as defined by Adams, the two fluid‐dynamical equations are derived in the three‐dimensional space. This generates a “classical” view of such quantum systems, corresponding to an MCSCF wave function: The many‐electron Schrödinger fluid consists of individual fluid components, each corresponding to a natural orbital and having its own charge density and current density. The local observables, viz., the net charge density and net current density, are obtained by merely summing over the natural orbitals, with the occupation numbers as weight factors; but, the net velocity field cannot be so obtained. Further, although each fluid component moves irrotationally in the absence of a magnetic field, the net velocity field is not irrotaional. The irrotational character of each velocity component is destroyed by rotation of the nuclear framework of the system while electron spin introduces an additional term, the spin magnetization moment, into each component current density. The physical significance of the fluid‐dynamical equations as well as their advantages and disadvantages are discussed.

Ab initiocalculations of radiative transition probabilities in theX2Σ+andA2Π electronic states of CO+

Electric dipole moment and electronic transition moment functions have been calculated for the X 2Σ+ and A 2Π states of CO+ using MCSCF wavefunctions and flexible basis sets. From these data and RKR potential energy functions, transition probability coefficients of spontaneous emission have been evaluated for radiative transitions within and between the X and A states. The calculated lifetimes of the A 2Π state (v′=0–9) are in good agreement with experimental values. Due to a strong curvature of the dipole moment function near the equilibrium distance, intense infrared overtone transitions are predicted in the X 2Σ+ state. It is shown that high vibrational levels (v≥15) of the X 2Σ+ state can be effectively depopulated via electronic transitions to energetically lower lying vibrational levels of the A 2Π state.

Multiple excitations and charge transfer in the ESCA N1s (NO2) spectrum of paranitroaniline. A theoretical and experimental study

The N1s (NO2) ESCA spectrum of paranitroaniline (PNA) is analyzed theoretically and experimentally. New monochromatized high-resolution spectra in the vapor (gas) and condensed phases are presented. In the condensed phase spectrum two distinct peaks are discerned, whereas the gas phase spectrum shows a drastically different feature in which the two peaks are reduced to an asymmetric band. The gas phase spectrum has been interpreted based on large-scale MCSCF wave functions calculated using a complete configuration expansion in the π valence orbital space. Emphasis is placed on the identification of features due to ’’shake’’ effects. Multiple excitations in the charge transfer excited states, as well as configuration near-degeneracy effects in the NO2 group, are found to be crucial for this identification. Our results are different from an earlier many-body, Green’s function, treatment of PNA. We find that the charge transfer shake states lie higher in energy (shakeup) and not lower (negative shakeup).

A b i n i t i o calculations of low lying states of the BH+ and AlH+ ions

For the X 2Σ+, A 2Π, and B′ 2Σ+ states of the BH+ and AlH+ ions potential energy, electric dipole and transition dipole moment functions have been calculated from MC–SCF wave functions. For the X and A states the MC–SCF results are compared with those obtained from highly correlated PNO–CEPA wave functions. All emission processes in the X, A, and B′ states have been investigated. The absolute emission intensities for the most intense bandheads of the A→X and B→X transitions have been calculated, and the band shapes are compared with the experimental emission spectrum of BH+ obtained in the reaction B+ (1–10 eV/cm)+H→BH++H. A so far unobserved part of the B′–X emission system is predicted between 2000 and 2500 Å. The Franck–Condon factors and the energies for the ionizations of the AH molecules into the AH+ ions are given.

Ab initio calculations of infrared transition rates in the ground states of BF and BF +

Electric dipole moment functions and radiative transition rates have been calculated for the X 1Σ + state of BF and the X 2Σ + state of BF + from MC SCF wavefunctions. Both are predicted to be strong emitters in the infrared. For BF + the electronically excited 2Σ + state, corresponding to the configuration 1σ 22σ 23σ 24σ5σ 21π 4, is calculated to be bound, whereas the A Π state is repulsive.

Theoretical characterization of negative ions. Calculation of the electron affinities of carbon, oxygen, and fluorine

The electron affinities of carbon, oxygen, and fluorine have been calculated using a compact MCSCF wave function with a [4s, 4p, 3d] Gaussian basis set. This wave function describes radial correlation of the 2p electrons which is found to have a large differential effect between atom and anion, and also includes the (2s2→2p2) near-degeneracy effect. Radial correlation of the 2p electrons increases the calculated electron affinity by as much as 1.2 eV over the Hartree–Fock value. An orbital model is discussed, which ascribes this effect to the diffuse nature of the orbital occupied by the (Z+1)st electron of the anion. Configuration interaction calculations based upon the MCSCF wave function result in electron affinities comparable in magnitude to the large basis set calculations of Yoshimine and Sasaki [Phys. Rev. A 9, 17 (1974)]. Use of the MCSCF wave function as the reference function for the CI calculations efficiently includes the effect of the triple and quadrupole excitations (relative to the HF wave function) found to be important by Yoshimine and Sasaki.

Valence correlation in the s2d n, s d n+1, and d n+2 states of the first-row transition metal atoms

The major differential valence correlation effects of the lowest lying states arising from the s2dn, sdn+1, and dn+2 configurations of the first-row transition metal atoms have been characterized using MCSCF and CI procedures. The important correlation effects are found to be, first, angular correlation of the 4s2 pair arising because of the near degeneracy of the 4s and 4p orbitals and, second, radial correlation of the 3d electron pairs. This large differential radial correlation of the 3d electrons can be interpreted as being due to nonequivalent d orbitals in the sdn+1 and dn+2 excited states. Both of these effects can be incorporated into a simple MCSCF wave function that reduces the error in the excited state atomic dissociation limits (∼0.2 eV in Sc–Cr and ∼0.5 eV in Mn–Cu for the sdn+1–s2dn excitation energy), yet still is of a form which lends itself easily to molecular calculations.

MCSCF calculation of the dipole moment function of CO

Potential energy and dipole moment functions for the ground state of CO have been calculated using MCSCF wavefunctions. Some difficulties with the MCSCF method at large distances, which are due to the near degeneracy of the two asymptotic states, are discussed. From the theoretical dipole moment function and the RKR potential function vibrational dipole matrix elements have been evaluated. For the fundamental and first overtone sequences, the theoretical values are in excellent agreement with experimental data. For transitions involving large vibrational quantum numbers (ν″ > 25) the calculated matrix elements are expected to be the most reliable ones presently available.

A second order MCSCF method for large CI expansions

A second order MCSCF method is presented in a form which is suitable for large CI expansions. This methodology allows one to employ second order MCSCF theory to problems in which the CI expansion is an order of magnitude larger than could be addressed previously. Sample calculations are reported on the X 1Σ+ and 2 1Σ+ states of HF in which the MCSCF wave function was composed of 1436 CSFs. (AIP)

MCSCF study of the avoided curve crossing of the two lowest 1Σ+ states of LiF

Potential energy, dipole moment, and transition moment functions have been calculated for the two lowest 1Σ+ states of LiF using MCSCF wave functions. By optimizing both states simultaneously with a particular choice of configurations it is possible to account for the major part of the correlation energy difference of the neutral and ionic states and to obtain the correct separation of the states at infinite internuclear distance. A simple method to obtain a diabatic description of the two states is proposed. In the crossing region nonadiabatic coupling elements have been calculated.

A hybrid method for improving MCSCF convergence

It has been found that the convergence problems for many ill conditioned single-configuration SCF calculations arise from mixing among only a small number of orbitals. This orbital set includes the highest closed, the partially filled, and (possibly) a few of the lowest virtual orbitals. The fact that convergence problems can be, in very large measure, linked to a small orbital set is used to design a hybird MCSCF procedure in which these orbitals are treated using a second-order MCSCF method, while other mixings are treated with a lower-order method which avoids the time consuming integral transformation. Tests on BeO show that the hybrid method yields convergence even when the simple lower-order treatment diverges. The method is expected to facilitate determination of MCSCF wave functions for large basis problems and for the construction of potential energy surfaces.

An SCF and MCSCF description of the low-lying states of MgO

SCF and MCSCF wave functions for the low-lying 3,1Σ+ and 3,1Π states of MgO are presented and discussed. It is argued that a viable description of these states is possible if the 3Σ+ and 3,1Π states are treated at the SCF level while the 1- and 2- 1Σ+ states are treated with a limited MCSCF expansion.

MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem

AbstractA novel approach is developed for optimizing molecular orbitals within the context of a multiconfiguration self‐consistent‐field problem. The MCSCF wave function is determined through a sequence of eigenvalue problems in the multiconfiguration space and the single‐excitation space. They are used to iteratively improve the natural orbitals, which in turn are related, by successively improved transformations, to the MCSCF orbitals. The mathematical problems arising out of this general concept are solved and the computational implementation is discussed. In many applications the method has proven itself as a powerful approach in forcing rapid convergence. Adaptation to spin and spatial symmetry is maintained throughout and the procedure is applicable to excited states as well as to ground states.

Energy gradient in a multi-configurational SCF formalism and its application to geometry optimization of trimethylene diradicals

Previous methods of analytic evaluation of the potential energy gradient with respect to nuclear coordinates have been limited to single determinant Hartree-Fock wavefunctions. An expression and a practical method are presented for analytic gradient evaluation for MCSCF wavefunctions of certain widely applicable types. A geometry optimized of trimethylene diradical reaction intermediates has been actually accomplished by the use of the energy gradient for an ab initio MCSCF wavefunction.

An ab initio potential energy surface for the reaction between hydrogen fluoride and the hydride ion

A collinear potential energy surface for the reaction between hydrogen fluoride and the hydride ion is obtained using MC SCF wavefunctions. The form of the calculated surface in the entrance and exit channels is compared to that obtained using multipole expansions. The location of an avoided crossing on the surface is discussed.

Direct minimization of the energy functional in the LCAO-MO density matrix formalism

A previously proposed method of energy minimization is developed for MC SCF wavefunctions formed by all-pair excitations for a closed-shell system. The orbital coefficients are optimized by a gradient approach using a suitable orthogonal transformation of the atomic basis, while optimum CI coefficients are determined solving the usual secular problem for the lowest eigenvalue, after each optimization of the orbitals. Applications to LiH and NH3 molecules show that the method is numerically well stable, and is capable of accounting for a large part of the correlation energy giving results which compare well with those of the conventional CI method.

Charge distribution analysis in terms of Berlin's binding and antibinding regions for Li2 and F2

AbstractThe MCSCF wave functions of Das for Li2, and Das and Wahl for F2, have been analyzed in terms of Berlin's binding and antibinding regions. The effect of electronic correlation on the charge distributions in these regions is studied. The process of bond formation in going from separated atoms to the correlated molecule is also discussed with reference to Berlin's regions.

MCSCF calculations of the properties of hydrogen fluoride

Various properties of hydrogen fluoride are calculated as functions of internuclear separation using an MCSCF wavefunction. The properties considered are the dipole, quadrupole and octopole moments, the diamagnetic contributions to the magnetizability and the nuclear magnetic shielding, the electric field gradients at the nuclei, the polarizability and the A tensor. Vibrational corrections to these properties have been included. Good agreement with the experimental results is found for the whole range of properties.

Accurate a b i n i t i o potential curves for the X 2Πg, A 2Πu, a 4Σ−u, and 2Σ−u states of the 0−2 ion

A b initio potential energy curves and spectrocopic constants of the X 2Πg, A 2Πu, a 4Σ−u and 2Σ−u of the negative ion of molecular oxygen (0−2), of major interest in the upper atmosphere, are presented in a sequence of increasing computational sophistication. The unique features of the correlation problem for homonuclear ions are analyzed in detail. The most accurate 76-configuration MCSCF ground X 2Πg state results are examined and found (as expected) to be in excellent agreement with experiment. The most accurate A 2Πu excited state results (also involving a 76-configuration MCSCF wavefunction) are expected to be of comparable accuracy, while somewhat less accuracy is expected for the best calculations of the 4Σ−u and 2Σ−u states (64 and 65 configurations used, respectively).

MCSCF wavefunctions for the a 1Δg and b 1Σ+g states of O2

MCSCF wavefunctions for the <i>a</i> 1Δ<i>g</i> and <i>b</i> 1Σ+<i>g</i> states of O2