Symmetry adapted versus symmetry broken wavefunctions: the 1s core level ions of O+2

It is shown that, for O₂, in a MC SCF determination of the core ionization potentials employing the full Molecular point group, very few (N-1)-particle configurations are required in order to account for the symmetry breaking in the corresponding Hartree-Fock calculations.     

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.     

Ab initio calculations on porphin

Ab initio SCF calculations are reported for the porphin molecule. The positions of the central protons have been optimized, and the equilibrium geometry is found to be a linear NH ? HN arrangement. The NH vibrational frequencies have been computed and are compared to experimentally measured quantities. Several low ionized states have also been studied in separate spin-restricted SCF calculations. The lowest state is found to have B_1u symmetry with an ionization potential of 8.0 eV.     

Broken orbital symmetry and the description of valence hole states in the tetrahedral [CrO4]2− anion

The localization of ligand-based valence holes in the tetrahedral complex ion [CrO₄]^2? in a crystalline environment is studied by SCF calculations on the hole states, with progressively less restrictions on the spatial symmetry of the molecular orbitals. The final wavefunctions are obtained by constructing, from the symmetry broken SCF solutions, wavefunctions that exhibit again the proper transformation properties under the operations of T _d . The crystal environment of the [CrO₄]^2? anion is represented by a point charge model. In contrast with the situation for core hole states, the projection afterwards into T _d symmetry is important. The final ionization energies, which are obtained from projected C _3v adapted SCF solutions, are reduced considerably (?3 eV) with respect to the T _d ΔSCF results, but the ordering of the states has not changed essentially. The calculated ionization energies compare favourably with results of XPS experiments on Na₂CrO₄. The evaluation of the energies of projected symmetry broken SCF solutions requires the calculation of hamiltonian matrix elements between determinantal wavefunctions built from mutually non-orthogonal orbital sets. An efficient method for the calculation of such matrix elements is presented.     

SCF and limited CI calculations for assignment of the Auger spectrum and of the satellites in the soft X-ray spectrum of H2O

SCF and limited CI calculations have been performed for a number of excited states of doubly ionized water. The calculations allowed an assignment of both the Auger spectrum and most of the satellites in the soft X-ray spectrum. The assignment for the two highest peaks in the Auger spectrum was ambiguous. SCF calculations of the K emission spectrum of H₂O were also performed. It was found that limited CI calculations were of importance for some states with two open shells in one symmetry. Such effects are caused by large off-diagonal lagrangian multiplies connecting the open shells in the reference configuration.     

A theoretical study of the isotope separation of Li2 using different laser wavelengths for the excitation and ionization processes

Cross sections of the sequential two-photon ionization of Li₂ are evaluated starting from ab initio CI wavefunctions and using different laser wavelengths for the excitation and ionization processes. In agreement with experimental results and previous calculations, a substantial isotope separation is found for the three isotopes of Li₂, and a wide spectrum of wavelengths useful for effective isotope enrichment is suggested. Even in with an overcrowded spectrum, the fundamental role of Franck-Condon factors and thermal populations in defining the characteristic resonances of the fractionation process is confirmed. This result suggests a practical way to predict transitions useful for isotope enrichment.     

Broken orbital-symmetry and the description of hole states in the tetrahedral [CrO4]− anion. I. Introductory considerations and calculations on oxygen 1s hole states

The localization of holes in systems containing spatially equivalent sites is discussed in terms of a simple one-particle model in which quantum mechanical delocalization effects compete with essentially classical polarization or dielectric relaxation effects. The predictions of the model for a tetrahedral system like CrO^?₄ compare favourably with the results of symmetry unrestricted SCF calculations on O_1s hole states. The connection with a Cl treatment using symmetry-restricted MOs is discussed. The calculated ionization energies are finally compared with XPS measurements on Na₂CrO₄. To this end the crystal surrounding of the CrO^?₄ anion has been represented by a point charge model and the ensuing Modelung field was included in the SCF calculations. In contrast to the Td restricted result of 551.4 eV, the completely localized C_3v results of 532.6 eV is in satisfactory agreement with the experimental data which are found around 530.0 eV.     

Generalized-molecular-orbital theory: Simple multiconfiguration self-consistent-field method

Even after completing a multiconfiguration self-consistent-field (MCSCF ) calculation, one must often include additional configuration interaction (CI ) to obtain quantitative or semiquantitative results. There is some question of whether the prior MCSCF calculation is worthwhile, if additional CI is needed later. We have developed a new MCSCF computational method, which, because of our assumptions about the nature of the configurations, yields one Fock-like operator for all the “filled” orbitals (high occupation numbers) and a second Fock-like operator for all the “virtual” orbitals (low occupation numbers). Since there are only two matrices to build, our method is considerably faster than other MCSCF approaches. Because of these similarities to standard molecular-orbital (MO ) calculations, we have termed our approach generalized-molecular-orbital (GMO ) theory. However, the “virtual” orbitals, unlike those of standard MO theory, are optimized to correlate the “filled” ones and can he used in a subsequent CI calculation. Results are presented for the correlation energy of H₂O, the spectroscopic constants of N₂, the singlet–triplet energy separations in CH₂, and the nature of the chromium–chromium quadruple bond. Although these results are at a very low level of CI , the GMO approach appears to correct for the gross deficiencies of the single-determinant SCF procedure.     

Experimental and theoretical investigation of the Li is spectra of molecular lithium halides

The Li 1s absorption spectra of molecular LiF and LiCl have been studied for the first time by experimental as well as by theoretical methods. The strong absorption bands at the onset of the Li 1s excitations can qualitatively be interpreted in terms of a simple ionic model. The spectra are analyzed and assigned in more detail on the basis of SCF and Cl calculations, and also within the framework of the Z + 1 core analogy model. The limitations of this model are discussed. Li 1s ionization potentials of LiF, Li₂F₂, LiCl, and Li₂Cl₂ are derived.     

Symmetry and equivalence restrictions in electronic structure calculations

A simple method for obtaining MCSCF orbitals and CI natural orbitals adapted to degenerate point groups, with full symmetry and equivalence restrictions, is described. Among several advantages accruing from this method are the ability to perform atomic SCF calculations on states for which the SCF energy expression cannot be written in terms of Coulomb and exchange integrals over real orbitals, and the generation of symmetry-adapted atomic natural orbitals for use in a recently proposed method for basis set contraction.     

Analysis of the space correlation factors in a MCSCF representation of LiH and Li2 molecules

The space correlation factor is studied in LiH and Li₂ molecules, using MCSCF wave-functions. The shape of the Fermi hole is related to the localization of molecular space orbitals, while the Coulomb hole study indicates the importance of symmetry properties of the molecular orbitals involved in excited configurations for the representation of the electronic correlation inside the chemical bond.     

Usefulness of modified virtual orbitals in multireference CI procedure illustrated by calculations on lithium clusters

The maximization of the exchange interaction between the canonical Hartree–Fock virtual and occupied orbitals leads to a transformed set of virtual orbitals which are well suited as one-electron functions for CI calculations. The procedure, generally known for a long time is seldom applied, despite its simplicity and very low computational demand. However, it is found to be particularly useful in the case of multireference CI, since an improved energy is obtained with a considerable shortening of the CI expansion. Moreover, in the final CI wave function, several configurations appear with considerable weight, thus allowing an easy choice of additional configurations to be inserted in the definition of a new zero-order wave function. The efficiency of the computational procedure is discussed for the case of a Li₆ cluster of D_3h symmetry and for the NaCO and PdCO complexes. Results are reported for the relative stability of four different geometrical arrangements of the Li₆ cluster.     

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.     

Vibronic coupling and symmetry breaking in core electron ionization

It is shown that for highly symmetric molecules the ionization of a core electron leads quite generally to a lowering of the symmetry. The breaking of the symmetry is a consequence of the vibronic coupling between nearly degenerate core orbitals of different symmetry. The vibronic coupling leads to strong excitation of non-totally symmetric vibrational modes in addition to the usually observed excitation of totally symmetric modes. As an example, the vibrational structure of the Ols line of the CO₂ molecule is computed on the one-particle level.     

The ground-state potential curve for F2

The ground-state potential curve for F₂ has been obtained using large-scale MC SCF and CI methods. MC SCF curves were obtained with the CAS SCF method using a variety of sets of active orbitals. The main conclusion from the CAS SCF calculations is that the 2π_u orbital is important. CI curves were obtained using the contracted CI method. The largest calculations contained 312000 configurations proper spin and space (d_2h) symmetry. The main conclusions from the CI calculations are that the configuration XXX are important, otherwise errors in D_e of 0.3 eV and in r_e of 0.02 Å are found. The remaining errors at the CI level are 0.08 eV for D_e, 0.005 Å for r_e and less than 10 cm^?1 for the lowest vibrational levels.     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

Wellenmechanische Absolutrechnungen an Molekülen und Atomsystemen mit der SCF?MO?LC(LCGO) Methode. VIII. Das System (Li2H)+

The (Li₂H)⁺ has been investigated ab initio in the linear configuration, with the H atom in the middle of the system, for five different distances R_LiH, taking all six electrons into account, using the Allgemeines Programmsystem/SCF? MO? LC(LCGO) Verfahren. A bond distance R_LiH of 3.14 a.u., a total energy of ?15.289 a.u., and an ionization energy of 15.1 eV were found. Comparing the results of SCF investigations, the formation energy of (Li₂H)⁺ from LiH and Li⁺ was computed to be 59.7 kcal/mole (2.58 eV). Using the energy curve near the minimum, a force constant for the symmetric vibration of k = 0.13777 × 10⁶ dyn/cm and a frequency ω = 577.9 cm^?1 were found.     

Theoretical study of the vertical electron affinity and ionization potentials of C3

The electron affinity and first three ionization potentials of C₃ are calculated using the multiconfigurational SCF and configuration interaction methods and by Möller-Plesset perturbation theory. Whereas Koopmans' theorem and SCF calculations indicate that the first cation state is ²Π_u, upon inclusion of correlation effects both the ²Σ_u and ²Σ_g cation states are found to lie lower in energy. CI calculations indicate that the ground state (²Π_g) anion is stable by 1.74 eV. Allowing for the error in the calculated electron affinity of the carbon atom, C₃^? is estimated to be stable by 2.0 eV, in excellent agreement with the 2.05 eV value determined from recent photodetachment measurements. No excited anion states are found to be bound at the equilibrium geometry of the neutral molecule.