Excited Orbitals Research Articles

Fast calculation of excited-state potentials for rare-gas diatomic molecules: Ne 2 and Ar 2

A fast method for obtaining excited-state potentials of rare-gas diatomic molecules is described. Two types of excited orbitals are used: molecular orbitals calculated in the field of a singly charged molecular ion, and atomic orbitals (properly symmetrized) obtained in a similar atomic system. The RPA equations are solved within the manifold of excitations from the highest occupied orbital in each symmetry to the lowest excited orbital of either type in each symmetry. A simple model for estimating the dynamic correlation correction to excitation (and ionization) energies is given. Applications to excited states of Ne 2( 1,3Σ + g, u, 1,3Π g, u) and Ar 2( 1,3Σ + g, u) are described. Two-electron integral transformations involve only three orbitals of each symmetry, and the RPA matrices are four-dimensional. The computational effort required for all excited-state potentials adds less than one-tenths (in terms of computer time) to the effort involved in the preliminary ground state Hartree—Fock calculations. The resulting potentials compare favorably with more elaborate CI calculations and give good agreement with spectroscopic and scattering data. Potential curves for the molecular ions are also given.

Momentum eigenfunctions in the complex momentum plane. II. Relation of secondary singular points to one-electron, local, potential functions

In a previous paper [J. Chem. Phys. 64, 4375 (1976)] it was shown that the points P=±i√−2E in the complex momentum plane (E is the one-electron energy eigenvalue) are singular points of the one-electron momentum eigenfunction obtained from a one-electron local potential function. In the present research a whole series of additional (secondary) singular points are located. These depend on the specific nature of the one-electron potential function in particular the types of exponentials contained in the potential. For bound states these singular points (like the primary singular points) lie on the imaginary axis but are further from the origin than the points ±i√−2E. Moreover, the positions in the P plane of the primary singularites are independent of direction of the momenturm vector and the same is also true of the secondary singular points for each of the (independent particle) potential functions considered in the present research, a major simplification. Since the primary singular points are closer to the origin than any others, they determine the radius of convergence of power series expansions and also the transformations which provide analytic continuation of the power series. The new singular points do not affect the convergence of series expansions used in past research. Collision amplitudes, in Born approximation, representing elastic scattering of an electron by a static, local, potential field also possess primary singular points on the imaginary axis of q (q being the magnitude of a momentum difference vector q) and secondary singular points further out on the imaginary axis. (The amplitudes for inelastic scattering also possess such singular points.) Again, the secondary singularities do not affect the convergence of series expansions used in past research [e.g., J. Chem. Phys. 57, 4357 (1972)]. Moreover, the positions of the singular points in the q-plane are independent of the direction of q. In momentum representation the Hartree–Fock equations are integral equations with kernel functions which are collision amplitudes, in Born approximation, for elastic scattering with exchange. The exchange amplitudes arise from nonlocal potentials, but approximate local potentials can be obtained by replacing the exchange amplitudes by functions only of a difference vector q and applying an inverse Fourier transform. The collision amplitudes corresponding to several local potential functions proposed by other investigators are examined for primary singular points. It is found that the primary singular point in the q-plane is related to one-electron eigenvalues in the same way for each of the potentials and hence a general expansion of the corresponding amplitude is suggested. From the inverse Fourier transform a series is obtained for the local potential. If the coefficients of the series are adjusted to give minimum energy for the system then the variable terms of the series constitute a basis set for the representation of local potential functions. For atoms, two different basis sets are discussed, one leading to a linear combination of Yukawa functions. The nonlinear Hartree–Fock equations can be arranged as integral equations in two different ways which lead, with the same approximations, to two different local potential functions for the same filled orbitals. If the same potential functions are applied to obtain excited (virtual) orbitals one set corresponds to a negative ion and the other to a neutral atom. The conditions on the atomic orbitals used in LCAO treatments is discussed.

Valence and Rydberg excited states of H2S: An SCF-Xα-SW molecular orbital study

SCF-Xα-SW calculations have been performed for H2S in the ground state, in ionic states resulting from the vertical removal of a valence electron, and in 36 singlet and triplet excited states resulting from the vertical excitation of an electron from the 2b1 or the 5a1 molecular orbitals to various virtual orbitals. Excellent agreement for the transition energies is found both with the experimental ultraviolet and photoelectron spectra and with recent large-scale configuration interaction calculations. The average absolute difference between the Xα-SW and CI results for the 26 cases where both exist is only 0.24 eV. The intrinsic advantages of the Xα-SW method over the traditional LCAO or one-center approaches for the treatment of Rydberg states are delineated. Contour plots of the excited orbitals have been generated and these are used to illustrate some of the conceptual problems related to the Rydberg-valence distinction. Finally, a brief speculative discussion of the possible photochemical implications of the character of the two excited orbitals involved in the first absorption band is given.

Atomic multipole polarizabilities in the extended Coulomb approximation

A new and computationally convenient exact analytic expression for the 2l-pole static polarizabilities of mono and divalent atoms and ions is obtained within the extended Coulomb approximation of Adelman and Szabo. In this approximation the valence electron wavefunction is represented by an asymptotically valid Whittaker function with hydrogenic normalization and the l-symmetry excited orbitals are eigenfunctions of a Coulombic Hamiltonian plus an r−2-type pseudopotential. Our expression for the polarizability contains two parameters which can be obtained from the ionization potentials of the ground state and the first l-symmetry excited state of the system. Illustrative calculations for the alkali and alkaline earth atoms are present. Good agreement is found with recent experimental results as well as those from elaborate calculations using correlated wavefunctions, where such results are available.

On the photoelectron and ultraviolet spectra of molecules containing a divalent sulfur atom and a carbonyl group separated by a methylene group

Photoelectron spectra show that in CH3SCH2COCH3 the sulfur and carbonyl lone pair orbitals [Formula: see text] are both stabilized with respect to the compounds containing S or CO only. This is in keeping with the observed values of the basicity constants. The ultraviolet absorption spectra contain the expected [Formula: see text] and [Formula: see text] bands and give evidence for a significant amount of mixing between the excited orbitals.

Calculation of doubly excited energy levels of rare-gas atoms by a frozen-core superposition-of-configurations method. I. Description of method and comparison with experiment for optically allowed levels of Ne

A method is described for the calculation of doubly excited energy levels of rare-gas atoms. The calculation makes use of frozen-core Hartree-Fock orbitals, optimized orthogonalized Slater-type functions to describe excited orbitals and virtual orbitals appearing in a superposition of configurations, and an empirical correction for core correlation. The method is found to be accurate to 0.1 eV for the 1s22s22p4(3P,1D,1S)3snp(1Po) levels of neon (n=3,4), as well as the singly excited 1s22s2p6np(1Po) levels (n=3-6). Modifications are described for the case of two or more states of the same symmetry, formed from the same core state and the same excited orbitals (e.g. 1s22s22p4(3P)3s(2,4P)3p(3Po)).

Polarization CI wavefunctions: the valence states of the NH radical

The polarization configuration interaction (POL–CI) wavefunction is described. Calculations are reported for the X 3Σ−, a 1Δ, b 1Σ+, A 3Π, and c 1Π states of the NH molecules in a double zeta plus polarization Gaussian basis. Calculated excitation energies and spectroscopic constants are compared with results from more extensive calculations in this basis and with previously reported calculations. The starting point for POL–CI calculations is a generalized valence bond (GVB) calculation to insure proper dissociation. Results of using both ground state and excited state GVB orbitals show that a single set of (ground state) orbitals may be used for all states without loss of accuracy. Dipole moments and electronic transition moments as a function of R are also given.

Excited states of Li 2+

The potential curves of some excited electronic states of Li 2 + leading to dissociation fragments up to Li + (1s) 2 + Li *(1s) 2 (3d) have been calculated with a frozen core method for nuclear separations between 2 and 20 bohr. Six curves are found to be binding, the first four 2Σ g + and two 2Π u states; four further states show a tiny energy minimum at large distances. The results are discussed with regard to a comparison with H 2 +, binding properties and the Rydberg character of the excited orbitals.

Excitation of the autoionizing level in Zn I and Cd. I

Electron impact differential and integrated cross sections for the excitation of the autoionizing level in zinc (3d to 4p) and cadmium (4d to 5p) are calculated using the first Born approximation (FBA) and the modified first Born approximation (MFBA). The orbital approximation has been employed. The ground-state orbital (nd, n=3 for Zn and n=4 for Cd) is represented by analytical SCF orbitals as obtained by Clementi (1967). The excited orbitals (4p and 5p) have been approximated by hydrogenic orbitals with appropriate effective Z.

Effective valence shell interactions in carbon, nitrogen, and oxygen atoms

We present the first numerical results of the canonical tranformation formalism for generating the valence shell effective interactions. Considering the valence space to be formed as the antisymmetrized direct product of 2s and 2p type orbitals we have computed all the matrix elements of the two-body valence shell interactions for isolated carbon, nitrogen, and oxygen atoms. Explicit forms for the core, valence, and excited orbitals are the hydrogenlike functions defined by an effective nuclear charge. We compare our results with the semiempirical ’’electron–electron repulsion parameters’’ finding that for reasonable values of the effective charge—essentially a choice of the core and valence spaces—good agreement with the parameters deduced from atomic spectra data is obtained. Not only do the general features of semiempirical theory such as the reduction of the average strength of the two-body interaction appear, but also the somewhat more subtle effects such as the relative ordering of the diagonal matrix elements emerge. The nonlocal nature of the effective interactions is noted and its implications for the usual semiempirical schemes are discussed.

A b i n i t i o study of the π-electron states of t r a n s-butadiene

Extensive ab initio configuration interaction calculations were carried out on the π-electron states of trans-1,3-butadiene. A double-ζ contracted Gaussian basis set, augmented with two diffuse 2p π functions on each carbon atom, was used in the calculations, which were based on a frozen σ core obtained from a ground-state SCF calculation. All excitations fro and to π orbitals were included in the CI treatment. Natural orbitals were obtained for many of the wavefunctions,and their spatial extenty was determined. Only five of the calculated excited states were found to have a valencelike charge distribution (computed vertical excitation energies in eV are given in parentheses): 1 3Bu (3.45), 13Ag(5.04), 21Ag(6.77), 33Bu (8.08), and 15Ag (9.61). These states all correlate with the valence N and T states of ethylene and can be readily described in terms of the ’’two-vinyl model’’ as either N T (the first two) or TT (doubly excited, the last three). Except for the doubly excited 21Ag, all low-lying singlet excited states were found to involve diffuse excited orbitals (as in the case of similar calculations on ethylene). Their identification as V states and/or Rydberg states is not always clear-cut. The calculated vertical excitation energies for the NT states are in good agreement with electron impact data, but owing to limitations of the frozen σ core, computed excitation energies for the diffuse states (particularly those related to the ethylene V state) are too higy by as much as 1 eV. The absorptions in the butadiene spectrum at 5.9 and 7.1 eV are believed to be due to the 11Bu and 21Bu states, computed at 7.05 and 8.06 eV, respectively.

Continuous change from valence to Rydberg type states. An example of XUV spectroscopy

The 2p-excitation spectra in the XUV region of argon and of the isoelectronic second row hydride molecules HCI 1 H 2S 1 PH 3 and SiH 4, recently recorded by Hayes and Brown, are discussed and assigned on the absis of qualitative arguments and ab initio calculations. The excited orbitals and the corresponding electronically excited states change their character continuously in going from HCI over H 2S and PH 3 to SiH 4 and Ar. In HCI the lowest excited orbital is an antibonding σ* valence MO, according to its extention, its orbital energy ϵ its dϵ/d R value, and its topology and number of nodes. However, in SiH 4 ( r 2 and ϵ have values typical for Rydberg orbitals: the topology has changed and the number of nodes has increased by one, so that the lowest σ* orbital in SiH 4 is very similar to the 4s-Rydberg orbital of Ar. Correspondingly the 4s-Rydberg orbital of HCI correlates with the 5s-Rydberg orbitals of SiH 4 and Ar. There occurs a significant Rydberg s-p mixing in HCI and H 2S. All these changes can be rationalized in terms of overlap integrals.

Simplified methods for the ab-initio calculation of van der Waals interactions including exchange

It is demonstrated on the example of He 2 that the van der Waals energy can be calculated quite accurately by the multi-structure valence-bond method using a relatively simple “effective excited state” model. A simplified procedure for optimization of the excited state orbitals, based on work by Murrell et al., is described. The importance of exchange and charge penetration effects, particularly on the dispersion forces, is quantitatively studied.

First principles derivation of the effective valence shell Hamiltonian for large molecules

We construct an effective Hamiltonian for the valence shell of a large electronic system. The procedure begins by classifying the complete one electron space according to core, valence, and excited orbitals. An N−electron subspace 𝒯N spanned by Slater determinants with a fixed set of Nc core orbitals and different sets of Nv = (N − Nc) valence orbitals is defined. A canonical transformation on the Coulomb Hamiltonian is used to eliminate the interaction between 𝒯N and its orthogonal complement, ?N, thereby defining an effective N−electron Hamiltonian. This effective Hamiltonian is expanded in a cluster development of linked one−, two−, three−,⋅⋅⋅body operators in terms of which the conditions of having vanishing matrix elements between 𝒯N and ?N can be explicitly formulated. Then starting with this effective N−electron Hamiltonian we construct an equivalent Hamiltonian to operate in the space of antisymmetrized products of valence orbitals. To within a constant (times the identity on the valence space) this equivalent Hamiltonian—dubbed the valence shell Hamiltonian—has the same matrix elements between antisymmetrized products of valence orbitals as does the effective N−electron Hamiltonian on 𝒯N. This formalism is then applied to generate an effective pi−electron Hamiltonian for planar conjugated molecules. We conclude by deriving explicit formulas for the transformed dipole transition operators.

Calculation of the hyperfine interaction using an effective-operator form of many-body theory

The effective-operator form of many-body theory is reviewed and applied to the calculation of the hyperfine structure. Numerical results are given for the $2p$, $3p$, and $4p$ excited states of Li and the $3p$ state of Na. This is the first complete calculation of the hyperfine structure using an effective-operator form of perturbation theory. As in the Brueckner-Goldstone form of many-body theory, the various terms in the perturbation expansion are represented by Feynman diagrams which correspond to basic physical processes. The angular part of the perturbation diagrams are evaluated by taking advantage of the formal analogy between the Feynman diagrams and the angular-momentum diagrams, introduced by Jucys et al. The radial part of the diagrams is calculated by solving one- and two-particle equations for the particular linear combination of excited states that contribute to the Feynman diagrams. In this way all second- and third-order effects are accurately evaluated without explicitly constructing the excited orbitals. For the $2p$ state of Li our results are in agreement with the calculations of Nesbet and of Hameed and Foley. However, our quadrupole calculation disagrees with the work of Das and co-workers. The many-body results for Li and Na are compared with semiempirical methods for evaluating the quadrupole moment from the hyperfine interaction and a new quadrupole moment of $^{23}\mathrm{Na}$ is given.

Molecular Orbitals for Excited States

The use of Mulliken's "ionic Hamiltonian" to obtain excited state orbitals is investigated. When compared with the ordinary virtual orbital method it is found to give improved charge distributions for the excited orbitals, but seems to be of less value for improving excitation energies in a minimal-basis calculation. As examples, INDO calculations have been carried out on the ethylene, formaldehyde, and pyridine molecules.

ChemInform Abstract: ORBITAL REDUCTION FACTORS IN THE LOWEST EXCITED STATE OF THE PHTHALOCYANINE RING AND THEIR MEASUREMENT BY MAGNETIC CIRCULAR DICHROISM SPECTROSCOPY

Magnetic circular dichroism (m.c.d.) spectroscopy has been used to measure the angular momentum of the lowest excited state of the phthalocyanine ring as a function both of the central metal ion and of the axial ligands bound to the metal. The magnetic moments have been extracted from experiment using a moments analysis thus providing numbers which are independent of any distortions, static or vibronic, within the excited state. It is shown that on changing the metal ion from cobalt(II) to zinc(II) a gradual increase in angular momentum occurs whereas changing the axial ligand on iron(II) phthalocyanine from dimethyl sulphoxide to cyanide produces a large change in magnetic moment from 3.24 to 4.58 Bohr magnetons. A simple model has been proposed to account for these observations. It involves molecular orbital formation between the eg(dxz, yz) metal d-orbitals and the empty 6eg(π*) orbitals of the phthalocyanine ring. It is shown that this necessarily leads to a reduction of the orbital angular momentum. With simple assumptions it is possible to estimate the molecular orbital coefficients for the metal d-orbital participation in the excited orbitals for a series of iron(II) phthalocyanines with different axial ligands.

Variational method for excited hartree-fock functionals

In this article variational approach is considered for determining excited Hartree-Fock orbitals. It is shown that for a model of an excited state of a system with “frozen” framework the variational method is equivalent to the method of potential VN−1.

Orbital reduction factors in the lowest excited state of the phthalocyanine ring and their measurement by magnetic circular dichroism spectroscopy

Magnetic circular dichroism (m.c.d.) spectroscopy has been used to measure the angular momentum of the lowest excited state of the phthalocyanine ring as a function both of the central metal ion and of the axial ligands bound to the metal. The magnetic moments have been extracted from experiment using a moments analysis thus providing numbers which are independent of any distortions, static or vibronic, within the excited state. It is shown that on changing the metal ion from cobalt(II) to zinc(II) a gradual increase in angular momentum occurs whereas changing the axial ligand on iron(II) phthalocyanine from dimethyl sulphoxide to cyanide produces a large change in magnetic moment from 3.24 to 4.58 Bohr magnetons. A simple model has been proposed to account for these observations. It involves molecular orbital formation between the eg(dxz, yz) metal d-orbitals and the empty 6eg(π*) orbitals of the phthalocyanine ring. It is shown that this necessarily leads to a reduction of the orbital angular momentum. With simple assumptions it is possible to estimate the molecular orbital coefficients for the metal d-orbital participation in the excited orbitals for a series of iron(II) phthalocyanines with different axial ligands.

Structure of the hydrated electron

We study the short range interactions between an electron and the neighbor water molecules, with a modified version of the CNDO/2 method. The basis set used includes excited atomic orbitals which are fundamental in the study of the system (H2O)n−. Our calculations indicate that the electron is not associated to a vacancy or other type of defect but to a group of water molecules in the normal microscopic structure of ice and liquid water. In ice it can also be considered as an extra electron in an otherwise perfect crystal lattice (an electronic polaron). Our results are consistent with the hydration energy, spectra, and photobleaching experiments.