LCAO Approximation Research Articles

Self-Consistent-Field Theory. IV. LCAO Approximation for Excited States

The pseudo-eigenvalue equations previously obtained for any state of an electronic system are brought into the form required by the LCAO approximation. The matrix formulation involved is discussed and some practical methods for reducing the value of the maximum error of the evaluated energy are given.

On the Excess Electron and Hole Band Structures and Carrier Mobility in Naphthalene, Anthracene, and Several Polyphenyls

The excess electron and hole band structures of naphthalene, anthracene, and several polyphenyls have been calculated in the tight binding approximation. In addition the anisotropy and the pressure dependence of the mobility tensor has been calculated in the constant-free-time and the constant-free-path approximations. The molecular wavefunctions were represented in the LCAO approximation using Hückel coefficients with the carbon atomic orbitals represented by the best available Hartree—Fock SCF carbon 2p ground-state function involving a linear combination of four Slater-type functions. By this choice of atomic orbitals, we hope to account properly for the behavior of the wavefunction at the large internuclear distances relevant to this problem. All of the above compounds are characterized by a crystal structure containing two molecules per unit cell. There are, therefore, two bands for both the electron and the hole cases. The bandwidths are calculated to be of the order of 0.1 eV. In the a—1 and b—1 directions, the symmetric and antisymmetric bands are degenerate at the zone edge. The electron bands are appreciably split in the c—1 direction. Although there is no attempt to make absolute calculations of the mobility, the experimental data seem to be in adequate agreement with the scattering models considered. Inferences can be drawn which suggest that further understanding of the mobility will arise from a detailed investigation of the interaction of the charge carriers and the intermolecular and intramolecular vibrations.

Charge transfer mechanism of heterolytic aromatic substitution

The charge transfer mechanism of aromatic substitution presented by the present author and Tanaka in the previous paper is further developed from the viewpoint of the resonance interaction between no-bond and charge-transfer structures. According to the present mechanism which includes symmetrical charge transfer complexes of σ-type as stable intermediates, the reaction process can be divided into two steps, first of which is the formation of the σ-type complex and the second, the decomposition of the complex into final products: moreover, these two steps are shown to be interpreted as transfer processes between the non-bond and charge-transfer structures. Approximate formulae for the activation energy are derived. They show that the orientation of substitution is determined by localization energies of aromatic cations and anions for electrophilic and nucleophilic substitution, respectively. These quantities are actually evaluated with some aromatic hydrocarbons by a simple LCAO approximation. The calculated localization energies can explain satisfactorily the observed orientation rule. Furthermore, the necessary condition for the occurrence of aromatic substitution is deduced with regard to ionization potentials and electron affinities of aromatic substrate molecules and reagents, and also with regard to binding energies between two components in intermediate σ-complexes.

Theory of NMR Chemical Shifts in Conjugated Fluorine Compounds

Ramsey's equation for the nuclear magnetic resonance screening constant is analyzed employing a Hartree—Fock ground-state wavefunction. Using the LCAO approximation, a general expression is obtained which relates the screening parameter to localized charge distribution quantities (charge and bond orders). This expression is equivalent to one given by Karplus and Das. Pertinent two-center integrals over atomic orbitals are evaluated, employing, in part, the Fourier integral folding theorem. The relative magnitudes of the integrals over atomic orbitals makes possible a simplification of the general equation for F19 shielding to a form involving quantities associated with the fluorine atom and the adjacent carbon atom. Under the assumption of an invariant sigma framework, the F19 chemical shift between two similar conjugated fluorine compounds is shown to be determined chiefly by the values of the fluorine pi-electron charge densities and the carbon—fluorine pi-electron bond orders. The latter terms are almost as important as the former. The groundwork is laid for applying the derived equations, in a subsequent paper, to several systems of aromatic fluorine compounds.

Homopolar Dipole

Numerical tables for the homopolar dipole or overlap moment μh.d. associated with various molecular orbitals (MO's) are presented. The present tables include values of μh.d./R for the common MO's constructed from linear combinations of atomic orbitals (the LCAO approximation), one of which is an 1s, 2s, 2pσ, 2pπ, 3s, or 3pσ orbital. The calculations employ approximate atomic orbitals of the Slater type. Formulas for x̄AB, the x coordinate of the centroid of negative charge for an electron in an MO, are given as functions of the parameters p=12R(ζA+ζB) and t=(ζA−ζB)/(ζA+ζB), where R is the internuclear distance, ζ (n—δ) is the effective nuclear charge found by the Slater rules, and (n—δ) is an effective quantum number. Values of μh.d./R are also given for some MO's in which one AO is a hybrid. The variation of μh.d. with the parameters p and t is shown graphically for certain MO's. The overlap moment μh.d. has been computed for various chemical bonds by use of the tables. Overlap integrals for the AO combinations 1s—3dσ, 2s—5s, 2s—5pσ, 2pπ—3dπ, and 3s—3s were computed since these were not available in the literature over the necessary range of the variables.

Molecular Model of the Heisenberg Exchange Interaction

The electronic wave function of ${\mathrm{N}}_{2}$ has been calculated for a series of internuclear distances, in the simplest LCAO approximation, including the principal effects of configuration interaction. As the internuclear distance increases there is a well-defined sequence of regions in which the ground state is most closely approximated by configurations in which successively more orbitals are represented as localized functions (definitely associated with one of the two atoms) rather than as the odd or even linear combinations of these appropriate to the full molecular symmetry. This corresponds to a continuous change in the nature of the unrestricted Hartree-Fock valence orbitals from molecular to atomic character as the atoms are separated. In the intermediate range of internuclear separation, it is a better approximation to treat some of the valence orbitals as modified atomic orbitals, coupled by an antiferromagnetic Heisenberg exchange interaction, than as molecular orbitals. Various contributions to the Heisenberg "exchange integral," of the kinds considered for the transition metals, are evaluated and compared. It is found that the ordinary direct exchange (which leads to ferromagnetic coupling) is small compared with the sum of the various antiferromagnetic effects, none of which can be described within the traditional Hartree-Fock approximation, which for solids becomes the energy band theory. A method is proposed by which the magnetic interaction in solids could be evaluated quantitatively, by modifying the usual energy band calculations in the same way that the usual molecular orbital theory is modified in the present work. Similar refinements to the band theory of the transition metals have recently been proposed by Goodenough on empirical grounds.

Hyperfine Interaction and Quadrupole Coupling Constant in the Hydrogen Molecule Ion

An accurate calculation has been made of the interaction between the electron spin magnetic moment and a nuclear moment for the ground state of the hydrogen molecule ion. An accurate value of the electric field gradient at the nucleus was also obtained. The values are compared with values from an approximate LCAO wave function. The possibility of observing the paramagnetic resonance spectrum of this molecule is discussed.

Study of the Wurtzite-Type Binary Compounds. III. Valence Band Structure of BeO

The LCAO approximation is applied toward a description of the valence band in BeO, in which only the closed shell Be(1s)2O(2p)6 configuration is considered. Thus no account is taken of the deformation of the oxygen valence electron distribution by the polarization field along the c axis of the wurtzite-type crystal. The problem is solved along several lines of symmetry in the hexagonal brillouin zone. We find a band width of 18 ev with a depth of approximately 7 ev.

The Molecular Orbital Theory of the Interaction between an Atom and a Crystal Surface

The LCAO method of finding approximate one-electron wave functions is applied to an atom and crystal in mutual interaction. The case of a hydrogen-like atom and a crystal whose valence electrons occupy a band of states derived, in the LCAO approximation, from a single atomic state is discussed with the help of the simplifying assumptions of the tight binding approximation. Formulae are derived for the wave functions and energy levels of this system. If the interaction extends only over a group Ω of crystal atoms near the foreign atom then, depending on the values of the interaction parameters, there may be localized states associated with the foreign atom and the group Ω. The energies of these states lie outside the normal band (including any surface band) of crystal states and their existence allows for the formation of localized bonds between the atom and the crystal.

On the Orbital Approach to the Many-Electron Problem in Molecule

The lowest 3;E,.+ and I;E,.+ states of the hydrogen molecule, the interaction energy between two normal helium atoms and the 3B1,. and 1B1,. states of the ethylene molecule are treated by the method of the orbitals in the single configuration approximation. The results are encouraging and the reason for this success is that we have used orbitals which are a little more flexible than the conventional LCAO MO's. It is confirmed that the failure of the conv.entional ASMO LCAO approximation does not necessarily imply the break-down of the useful concept of molecular orbital itself.

The Oscillator Strengths of the 1s -2p , 2p -2s and 2p -3d Transitions of HeH2+

The oscillator strengths of the 1sσ-2pπ, 2pπ-2sσ and 2pπ-3dσ transitions of HeH2+ are calculated from the exact two-centre wave functions. A comparison is made with results obtained from the LCAO approximation with and without mixing, both the dipole length and dipole velocity formulae being used in each case.

The Oscillator Strengths of the 1s -2s , 1s -3d and 2s -3d Transitions of HeH2+

The oscillator strengths of the 1sσ-2sσ, 1sσ-3dσ and 2sσ-3dσ transitions of HeH2+ are calculated from the exact two-centre wave functions. A comparison is made with results obtained from the LCAO approximation with and without mixing, both the dipole length and dipole velocity formulae being used in each case.

A Semi-Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I.

A semi-empirical theory is outlined which is designed for the correlation and prediction of the wavelengths and intensities of the first main visible or ultraviolet absorption bands and other properties of complex unsaturated molecules, and preliminary application of the theory is made to ethylene and benzene. The theory is formulated in the language of the purely theoretical method of antisymmetrized products of molecular orbitals (in LCAO approximation), including configuration interaction, but departs from this theory in several essential respects. First, atomic orbital integrals involving the core Hamiltonian are expressed in terms of quantities which may be regarded as semi-empirical. Second, an approximation of zero differential overlap is employed, and an optional uniformly charged sphere representation of atomic π-orbitals is introduced, which greatly simplify the evaluation of electronic repulsion integrals and make applications to complex molecules containing heteroatoms relatively simple. Finally, although the theory starts from the π-electron approximation, in which the unsaturation electrons are treated apart from the rest, provision is included for the adjustment of the σ-electrons to the π-electron distribution in a way which does not complicate the mathematics. Electronic energy levels in the theory are expressed in terms of ionization potentials of atoms, resonance integrals of bonds, Coulomb repulsion integrals between two π-electrons on the same atom and between two π-electrons on different atoms, and penetration integrals between π-electrons and neutral atoms. Preliminary applications to ethylene and benzene in which only the carbon-carbon resonance integral is treated as an empirical quantity show that the theory can reproduce the results of the purely theoretical method with very little labor. The reasonableness of considering all of the above quantities as semi-empirical is pointed out, however, and it is through a detailed examination and exploitation of this in the second paper of this series that correction for the inadequacies of the π-electron approximation is made and improved agreement with experiment is attained.

LCAO Molecular Orbital Computation of Resonance Energies of Benzene and Butadiene, with General Analysis of Theoretical Versus Thermochemical Resonance Energies

The decrease in πx-electron energy for the change from a Kekulé to a proper benzene structure is computed purely theoretically by the method of antisymmetrized products of MO's (molecular orbitals), in LCAO approximation, using Slater 2pπx AO's (atomic orbitals) of effective charge 3.18, and assuming a carbon-carbon distance of 1.39A. The result (73.1 kcal/mole) is a theoretical value for the gross (vertical) resonance energy of benzene taken for constant C–C distances of 1.39A. In order to make a comparison with the net or ordinary empirical resonance energy, several corrections to the latter are required. The principal one is for the ``compression energy'' required to compress the single and stretch the double bonds of the Kekulé structure from normal single and double-bond distances to 1.39A. The others (not hitherto clearly recognized) involve hyperconjugation and related effects. The corrections are discussed and their magnitudes estimated, but a reliable value can be obtained only for the compression energy. Allowing for this alone, the computed net resonance energy is 36.5 kcal. This agrees, within the uncertainties due to the omitted correction terms, with the value (41.8 kcal) of the ``observed'' resonance energy Δ based on thermochemical data. Here Δ is the departure of the actual heat of formation ΔHf0 of benzene from the value given by a standard formula for nonresonating hydrocarbons. A new standard formula containing corrections for the mutual effects of neighboring carbon-carbon bonds is given; this is of interest also for itself, and its significance is briefly discussed. The analysis given in the paper serves to clarify hitherto existing obscurities in what is meant by ``resonance energy.'' An analysis of the nature and significance of ``nonresonating'' structures (like for example Kekulé benzene) is also included, using He2, 1,3 butadiene, and benzene as examples. Repulsion terms in the LCAO MO theory occur for nonresonating structures, which appear to be the counterpart of exchange repulsions of the valence-bond theory. When resonance occurs, it more or less overcomes these repulsions. The gross resonance energies of cis- and trans−1,3-butadiene are computed by the same method. For transbutadiene, the computed value corrected for compression is 3.7 kcal/mole, while the ``observed'' thermochemical value Δ is 6.5.

The Oscillator Strength of H2+, 1sσ—2pσ

The 1sσ—2pσ transition of H2+ is investigated using exact two center wave functions and using approximate wave functions based on a linear combination of atomic orbitals (with both the dipole length and the dipole velocity formulas). It is found that at the equilibrium internuclear distance, 2.00 a0, the true oscillator strength of the transition is 0.319; with the dipole length formula, the LCAO approximation yields a value some 1.4 times greater, and with the dipole velocity formula it yields a value some 0.54 times smaller. Results are given for internuclear distances between 0 and 9 a0.

LCAO Self-Consistent Field Calculation of the Twisting Frequency and π-Electron Energy Levels of Allene

The twisting frequency of allene is computed by the nonempirical method of antisymmetrized products of molecular orbitals in self-consistent field LCAO approximation, as applied to the four unsaturation electrons of the molecule, molecular orbitals for these electrons being taken as linear combinations of 2p Slater atomic orbitals. By comparison with a previous calculation by the same method on ethylene, the ratio of the twisting frequency of ethylene to the twisting frequency of allene is computed to be 1.7, sensibly independent of the effective charge in the atomic orbitals, which is the only extrageometrical parameter entering the theoretical calculations. The first ionization potential of allene is predicted to be 10.4 ev and the mean of the first allowed singlet electronic absorption band and the corresponding triplet is predicted to lie at 5.1 ev.

Molecular Orbital Calculations of the Lower Excited Electronic Levels of Benzene, Configuration Interaction Included

The lower excited π-electron levels of benzene are calculated by the non-empirical method of antisymmetrized products of molecular orbitals (in LCAO approximation) including configuration interaction. All configurations arising from excitation of one or two electrons from the most stable configuration are considered, and all many-center integrals are retained. The results are in better agreement with experiment and valence-bond calculations than those obtained previously by Craig in a calculation neglecting many-center integrals. Configuration interaction is found to change the order of the 1B1u and 1E2g states but leave unchanged the order of the 3B1u and 3B2u states, in agreement with the assignments 1A1g—3B1u and 1A1g—1E2g for the experimental bands at 3.8 and 6.2 ev.

Energy Levels and Color of Polymethine Dyes

The secular determinant for the energy levels of the unsaturation electrons of a polymethine dye is discussed in both HLSP and LCAO approximations. The roots of the secular determinant, which were obtained in the preceding paper by Herzfeld, are applied to a discussion of the longest wave-length electronic band of symmetrical and unsymmetrical polymethines. The LCAO approximation gives good numerical values for the dependence of both the transition energy and oscillator strength on the length of the polymethine chain in symmetrical dyes, but cannot handle the questions which depend sensitively on a small difference between the groups attached to the two nitrogen atoms at the extreme ends of the chain. Although the HLSP method yields correct qualitative results for symmetrical dyes, it does not give good numerical values. The HLSP method, however, is sensitive to a difference between the two ends of the dye molecule and affords a simple explanation for a number of properties in which unsymmetrical dyes differ from symmetrical ones.

Intensities of Electronic Transitions in Molecular Spectra IX. Calculations on the Long Wave-Length Halogen Spectra

The dipole strength DNQ for the perpendicular-type N→Q transitions in the long wave-length spectra of the halogen molecules is computed theoretically according to each of two approximations (LCAO and AO) for the wave functions. The work illustrates the behavior of antisymmetrical wave functions in transition moment calculations, and justifies the ignoring (as an approximation) of closed shells. The overlapping and dipole moment integrals ∫ [φ(npσ)][φ(npσ′)]dv and ∫ [φ(npσ)]y[φ(npπy′)]dv are evaluated using Slater's approximate atomic orbitals for the φ's. The calculated DNQ values for both approximations show the same rapid variation with atomic number as do the observed values, except for iodine, where the anomalous smallness of the observed value may reasonably be attributed to strong case c effects. The LCAO calculated values are consistently much larger than the AO values; the latter agree surprisingly well with the experimental values (cf. Table I). The calculations show DNQ to be very sensitive to variations of the internuclear distance r, and to variation of the effective charge Z* in the Slater atomic orbitals used. It is pointed out that the former variation may require reconsideration of work in which repulsive potential energy curves are deduced from intensity data on absorption continua. A qualitative discussion is given of the effects on DNQ of various possible improvements of the wave functions used. In connection with this, calculations have been made which show that the energy of the σu1s antibonding state of H2+ in LCAO approximation is minimized when Z* for 1s is (roughly) 0.84, for normal internuclear distance. This contrasts with Z*=1.228 determined by Finkelstein and Horowitz for the σg1s bonding state, and suggests that to improve molecular wave functions Z* should be increased for bonding but decreased for antibonding molecular orbitals.