Showing papers in "International Journal of Quantum Chemistry in 1974"

Configuration interaction calculations on the nitrogen molecule

Abstract: A contracted Gaussian basis set capable of describing about 63% of the correlation energy of N2 has been used in a detailed configuration-interaction calculation. Second-order perturbation theory overestimated the correlation energy by 23–50% depending on how H0 was chosen. Pair-pair interaction affects the correlation energy by about 20% while quadruple excitations have an 8% effect.

Perturbative ab initio calculations of intermolecular energies. I. Method

Abstract: Starting from a knowledge of approximate wave functions of the isolated molecules (or atoms) A and B, a method is proposed to build up a zeroth-order ground state and excited configurations for the complex AB in which the molecular orbitals keep their local significance. The standard Rayleigh-Schrodinger perturbation in this basis provides a decomposition of the zeroth-order interaction energy as a sum of the electrostatic and repulsion energy. In the second order, it is possible to identify the classical dispersion and polarization forces (modified by a term of order S2) and two additional contributions which are linked to the exchange possibility. The intramolecular correlation component is taken into account and compared with the correlation on the isolated molecules. It is moreover suggested that since we work in a rather limited basis set, the perturbed energy of AB must be compared with the unperturbed energies of A and B calculated in a basis including the vacant orbitals on their respective partner. Finally a possibility for going beyond the second order is described.

Correlation effects in the low–lying excited states of the PPP models of alternant hydrocarbons. I. Qualitative rules for the effect of limited configuration interaction

Abstract: Simple rules for an estimate of the correlation effects in the low-lying states of alternant hydrocarbons, as described by the Pariser–Parr–Pople Hamiltonian, are formulated These rules are based on the alternancy and spin symmetry classification of states in both strongly and weakly correlated limits and on the valence bond characteristics of those states in the fully correlated limit It is shown that the largest effect of the electron correlation will be found for the singlet “minus” states (using Pariser's classification of the alternancy symmetry species), a smaller effect for the triplet “plus” states, and a much smaller effect for the remaining states These rules are exemplified by limited CI calculations including all monoexcited and all mono- and bi-excited configurations, respectively, for a number of π-electronic systems In view of these rules the success of the PPP model in the monoexcited CI approximation may be understood

Ab initio calculations on porphin

Abstract: Ab initioSCF 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 B1u symmetry with an ionization potential of 8.0 eV.

Perturbative ab initio calculations of intermolecular energies. II. The He...He problem

Abstract: The ground state energy of the He2 system is calculated according to the techniques described in Paper I around the van der Waals equilibrium, without assuming the constancy of the intraatomic correlation energies. The second-order results do not present an attractive region; the S2 decrease of the intraatomic correlation correlation corrections is larger than the attractive interatomic correlation corrections. The further orders reverse progressively the situation and finally give a qualitatively correct potential curve after the fifth order. Reaching almost exact solutions in the considered basis, one can demonstrate that the intermolecular calculation of the system A + B in the union of the basis for A and B involves an energy decrease which simply represents the effect of the extension of the basis set; and that a proper intermolecular calculation must compare the A + B energy with the A and B energies calculated in conveniently extended basis sets including the vacant MO's of the partner.

Matrix elements in configuration interaction calculations

Abstract: A method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.

Half-projected and projected Hartree-Fock calculations for singlet ground states. II. Lithium hydride

Abstract: The half-projected Hartree–Fock function (HPHF) for singlet states is defined as a linear combination of two Slater determinants which contains only spin eigenstates with even spin quantum numbers. The possible uses of such an approach for determining molecular properties are investigated computing the potential energy curve, binding energy, force constant, and dipole moment variation corresponding to the lithium hydride ground state. Full projected and restricted Hartree–Fock calculations (PHF and RHF) are performed simultaneously for comparison purposes. It is found that the HPHF model yields very satisfactory results, very close to those of the PHF scheme. Both models predict properly the molecular behavior as a function of nuclear separation, whereas the RHF one fails. A discussion is given in terms of configuration equivalents. It is concluded that the HPHF scheme seems to be useful for determining molecular properties specially in the case of large systems in which the more sophisticated methods are unmanageable.

Ab initio calculations on urea

Abstract: Multiconfiguration wave functions constructed from contracted Gaussian-lobe functions have been found for the ground and valence-excited states of urea. ICSCF molecular orbitals of the excited states were used as the parent configurations for the CI calculations except for the 1A1(π π*) state. The 1A1(π π*) state used as its parent configuration an orthogonal linear combination of natural orbitals obtained from the second root of a three-configuration SCF calculation. The lowest excited states are predicted to be the n π π* and π π* triplet states. The lowest singlet state is predicted to be the n π π* state with an energy in good agreement with the one known UV band at 7.2 eV. The π π* singlet state is predicted to be about 1.9 eV higher, contrary to several previous assignments which assumed the lowest band was a π π* amide resonance band. The predicted ionization energy of 9.0 eV makes this and higher states autoionizing.

Static and dynamic first- and second-order properties by variational wave functions

Abstract: The problem of the evaluation of first- and second-order energies by the use of arbitrary variational wave functions is examined in detail for time-independent perturbations as well as for time-dependent perturbations. By using a compact formalism the general formulae to be used for the case of a fully optimized set of variational parameters are readily obtained and the most prominent features are examined. The generality of the approach is tested by showing how some widely used methods are obtained by using particular types of variational wave functions. The case of incompletely optimized sets of variational parameters is examined examined extensively and several approaches at different levels of approximation are proposed. Emphasis is put upon the importance of considering, in the calculation of higher-order energies, the variational parameters which may be of negligible importance, and thus often neglected, in the absence of perturbations.

Generalized open shell SCF theory

Abstract: A generalized form of the coupling operator technique in SCF theory has been developed. In the formalism presented here, the monoconfigurational problem may be treated as a particular case of the multiconfigurational framework. The matrix form of the operators has been analyzed; in the LCAO context a structure has been found which is very adequate for computational purposes. Some examples are also presented which show the usefulness of the theory, emphasising the CNDO and INDO approximations. Within the application of the method to ab initio calculations, some He and second row atoms states have been studied. The He first excited singlet is also studied, the result of the analysis of such a problem being that the nonorthogonality between the singlet functions of the fundamental and of the first excited states play a primordial role in the efficiency of the method. In no case have the calculation problems, appearing in the application of the theory, been of a more difficult nature than those normally found in the application of the formalism for closed shells.

Matrix elements for spin‐adapted configurations

Abstract: It is shown that the turn-over rule which is used to simplify matrix elements between spin-projected Slater determinants may also be used to simplify formation of matrix elements between any orthonormal set of spin-coupled Slater determinants. The coefficients for the spin-coupling may then be chosen freely to reduce the number of important configurations in the secular equation.

Perturbative ab initio calculations of intermolecular energies. III. The water dimer

Abstract: The interaction energy between two water molecules A and B is calculated by the method described in Paper I [1], previously applied for the interaction between two helium atoms (Paper II) [2]. This interaction energy is obtained as the difference between the energies of the complex (A + B) and the monomers (A) and (B), obtained by a perturbation method. The results obtained with the perturbation developed up to the second order in a minimal atomic basis set are decomposed into classical contributions and contributions linked to the exchange possibility. Charge transfer contributions are important and the localized character of the hydrogen bond is examined. It is pointed out that the definition of the set of excited configurations for the calculation of the energies of the isolated monomers is important, especially when one tries to use a small atomic basis set. A similar effect in SCF-type calculations is evaluated. The contribution of higher orders is evaluated by the CIPSI method.

The loge theory as a starting point for variational calculations. I. General formalism

Abstract: It is shown that any expectation value of any observable associated with a molecule is the sum of loge contributions and of loge pair contributions. This result provides a rigorous theoretical basis for the study of additive properties of molecules. It is demonstrated that molecular wave functions (exact or approximate) can be expressed as a sum of functions corresponding to the various electronic events. Furthermore any of these event functions can be expressed in terms of correlated loge functions. This expression suggests many kinds of variational procedures of calculating wave functions (known methods and new ones). The case in which noncorrelated completely localized loge functions are used is discussed. If continuous functions are used the variational equation reduces to a sum of independent variational equations, each one corresponding to a particular electronic event. This is not so when discontinuous functions are used or when a delocalized function is added to replace the correlation interloge function. The noncorrelated completely localized loge model is analyzed in more detail. It is seen that local spin operators can be introduced and that each event density operator is the product of the loge density operators. Therefore that model is an independent loge model. The corresponding generalized self-consistent field equations are derived. This treatment helps us to understand how a localized state of a molecule can produce an ion containing a delocalized region, a phenomenon which is sometimes at the origin of some misunderstanding in photoelectron spectroscopy. Finally it is seen how virtual loge functions can be introduced to describe excited states.

Matrix formulation of the generalized Hartree‐Fock methods

Abstract: Spin-projected one-particle density and spin density matrices are presented as polynomials of suitable unprojected quantities with generalized Sasaki-Ohno coefficients. Thus an explicit form of Harriman's theorems is given. For the two-particle spatial density matrix an expansion in direct products of powers of unprojected residual electron and spin density matrices is given. For these basic matrices of the scheme the variational spin-extended equations are formulated.

Method of complex molecular orbitals

Abstract: In the spirit of Lowdin's extended HF scheme, an SCF method for the description of the molecular ground state, based on complex molecular orbitals, is developed. As a special case the method of parity mixing in orbitals is formulated. A calculational procedure for solving derived secular equations is described and numerical results, obtained in the minimal basis set, are reported for the N2 and CO molecule. Resulting ground state energies are close to the energies of the CI calculations with the same set of orbitals.

The Coulomb hole in the 23S state of the helium isoelectronic sequence

Abstract: The pair distribution functions evaluated for the 23S state of the helium isoelectronic sequence from the Hart and Herzberg correlated wave functions and those corresponding to the Hartree-Fock approximation are used to determine the shape of the corresponding Coulomb holes. As a consequence of a discontinuity in the Hartree-Fock solution between He and Li+, the Coulomb hole has a different shape for He than for Li+ and the other isoelectronic ions.

Waller–Hartree spin‐free method

Abstract: In this paper we show that with the equivalent transformation Pr = (−1)P(Pσ)−1 the spin function dependent methods such as Slater's method without group theory or Goddard's method with group theory differ only in different antisymmetric requirements from the present Waller-Hartree spin function free method. There exists a one-to-one correspondence between Slater's determinantal wave function and the Waller–Hartree double determinantal wave function. Explicit expressions for the S2 operator, Lowdin's spin projector, matric basis and several different forms of spin-projected functions are given for the Waller–Hartree formalism. The results are compared with other methods including those of Slater, Matsen, Gallup, Goddard and Segal. The differences are quite significant. New spin operators are worked out using creation-destruction operators. A knowledge of group theory is not required in this Waller–Hartree method. We have also shown that the Waller–Hartree method is more convenient than Slater's method with spin functions especially in the evaluation of the functional ℋΨ/Ψ. The advantages and disadvantages in the use of a linear combination of N! Hartree products and linear combinations of all possible double determinants are discussed. In addition, a formula for the calculation of the Sanibel coefficients C(S, M, i) is obtained.

Matrix elements in all valence electron models

Abstract: Model Hamiltonians for all valence electron calculations are examined with regard to the possibility of obtaining consistent sets of matrix elements for the description of electromagnetic processes such as photon absorption, optical activity, and magneto-optical activity. Linear and angular momentum and position operator matrix elements are related to Hamiltonian matrix elements through symmetry relations and equations of motions. The possibility of elimination of empirical interatomic matrix elements is studied.

Equivalence‐restricted open‐shell SCF theory

Abstract: A method is presented for generating open-shell equivalence-restricted SCF orbitals in high-symmetry situations using Roothaan–Hartree–Fock programs which are adapted for lower symmetry.

Spin projected EHF method: Calculations for a four‐electron model system

Abstract: The spin projected extended Hartree-Fock equations for successive optimization of the orbitals are derived for the four-electron case and applied to butadiene in the PPP approximation. The method shows an excellent convergence and gives 91.4% of the correlation energy for the given model-Hamiltonian.

Nonbonded interactions in membrane active cyclic biopolymers. I. valinomycin‐potassium ion complex

Abstract: A method for quantitative comparison of various conformers of cyclic macromolecules in terms of nonbonded interaction energy terms is proposed. The first step in the procedure is an analytical evaluation of a set of Cartesian atomic coordinates consistent with the ring closure condition and prescribed values for internal parameters. A variant of the same technique enables different conformations of the macromolecule to be generated. Nonbonded interactions of the electrostatic, polarization and dispersion type and van der Waals repulsion are calculated using the generated coordinates, empirical bond polarizability data and results of simple molecular orbital calculations. An application of the proposed techniques to the closed symmetric conformation of valinomycin indicates that the macromolecule by itself is somewhat constrained in this state but that its affinity for a potassium ion is very strong—much stronger than the hydration energy of the latter.

Spin-orbit interaction in polyatomic molecules: Ab initio computations with Gaussian orbitals

Abstract: Standard sets of Gaussian atomic orbitals (STO-3G, STO-4.31G) are used to evaluate spin-orbit coupling constants in linear molecules (CO, NNN) and spin-orbit effects on singlet–triplet transition intensities in formaldehyde. All spin-other orbit effects have been included. In all cases spin-other orbit interactions form a large fraction of the matrix elements. Simple formulae to evaluate spin-orbit one- and two-electron integrals over atomic orbitals are presented. Standard molecular integral programs can be used for the computation of spin-orbit integrals.

Irreducible tensor operator methods in intermediate‐field coupling

Abstract: The algebra of irreducible tensor operators is developed in the intermediate-field coupling case. The Wigner-Eckart theorem is formulated for a simple irreducible tensor operator as well as for the Kronecker and scalar products of these operators. The expressions required for the calculation of Coulomb repulsion, crystal field splitting, spin-orbit interaction, and Zeeman effect are given in detail. Recent applications to various problems in spectroscopy and magnetism of transition metal compounds are referred to.

Coupled Hartree Fock calculation of static dipole polarizabilities and shielding factors of open shell systems

Abstract: Static dipole polarizabilities and shielding factors for the 2p open shell atomic systems are presented using coupled Hartree Fock theory in the framework of the Roothaan formalism. Calculations have been performed for the ground as well as for some valence excited states. A variational approach has been adopted for the determination of the first-order perturbed functions. The results are compared with those obtained from the correlated calculations and other techniques. The shielding factor values are in excellent agreement with the theoretical N/Z ratio.

On the need of precision in the calculation of the LCAO density matrix of polymers

Abstract: In this paper, we point out how the precision of the LCAO density matrix elements of polymers may affect the computed properties We propose to use a more adapted approach in order to evaluate these quantities

The self-consistent symmetrized OPW method with an application to crystalline selenium

Abstract: A review of the nonrelativistic self-consistent symmertrized orthogonalized-plane-wave (SCSOPW) method used for determining electronic energy bands in periodic solids is given. Working equations based on the full use of group theory at all stages are presented for elemental crystals. As an example the method is applied to the trigonal selenium crystal.

An improved Hellmann-type pseudopotential for atoms and molecules

Abstract: The concept of pseudopotentials offers much attractiveness for the quantum mechanical evaluation of the physical properties of atoms and molecules. The ideas of Hellmann, in which the repulsive and fermion character of inner electrons can be mimicked by an experimentally fitted, exponentially damped potential term, are especially attractive. Unfortunately, it is found that such a simple expression can only be used in a very limited number of cases, such as for the alkali metals, and even then fails for the simple case of lithium. The present study shows that the Hellmann idea can readily be extended by including a second “shielded potential” term evaluated from tabulated previous Hartree-Fock calculations. The new expression for the model pseudopotential is both simple and effective. With it, the inner potential of any of the alkali metal atoms, including lithium, can be represented so that calculation of the molecular properties of the metal dimers can be accomplished. Calculations for Li2, Na2, and K2 show the binding energies and equilibrium interatomic distances to be quite well given, in agreement with both chemical experience and spectroscopic evidence.

Potential energy integrals in semiempirical MO methods

Abstract: The two-center core-electron attraction integral VAB in zero-differential overlap semiempirical MO methods is examined. It is concluded that core-valence orthogonality and valence symmetrical orthogonalization effects must be considered, and that these effects provide justification for the CNDO/2 approximation VAB = ZBγAB.

Spin‐Projected extended Hartree–Fock equations. II. Odd‐electron systems

Abstract: The spin-projected extended Hartree–Fock equations discussed in Part I for an even number of electrons are given here for the odd-electron case.