Spectroscopic and Theoretical Aspects

BY R. DEVONSHIRE

1 Introduction

This Chapter covers the two-year period July 1975 to June 1977 inclusive. The format is broadly similar to that followed by David Phillips in previous years. My aim has been to make papers relatively easy to find, given that Specialist Periodical Reports do not contain a subject index. The selection of papers for the areas covered in this section is of necessity a very limited one; a simple listing of relevant papers would more than fill the available space. The end result is somewhere between being a telephone directory and a personal scrapbook.

A number of areas have developed significantly since this section last appeared and coverage of these is reasonably complete, viz. optoacoustic and related studies, single vibronic level studies, especially of the simple dicarbonyls, magnetic field effects, and laser photochemistry in general. No attempt has been made to report on the extensive literature on the photolysis of atomic or crystalline systems.

My overriding impression having completed this report of current work in a number of areas in photochemistry is of the subject's tremendous vitality at the present time.

2 Calculations of Electronically Excited States

The papers reported in this section are concerned with developments in the methods of calculation of spectroscopic and other related excited-state properties of atoms and molecules of interest in photochemistry. Although the coverage of calculations on particular systems is concentrated in this section, a number of papers containing extensive calculations appear in later sections.

An inequality formulation of the well known Hund's rules has been investigated numerically. Restricted Hartree–Fock SCF energies of various states of atoms and ions of the He, C, N, and O isoelectronic series were used to examine the behaviour of ΔE, Δ. the differences in total energy, electron–electron repulsion, and electron-nuclear repulsion respectively, in going from a singlet to a triplet state. ΔE was found to be positive in all cases because Δ was always positive and large enough to more than compensate for occasional negative values for Δ found, in particular, for the neutral atom cases. Viewed as a two-step process, the change from singlet to triplet involves firstly a conversion with frozen orbitals to an approximate triplet state which will certainly be at lower energy because of the smaller Vee. followed by a contraction of the wavefunction which results both in an increase in Vee, thus making it possible for an overall change in Vee, Δ, that is negative, and in a large decrease in Ven which makes Δ always positive.

A recent SCF theory of excited states has been used to calculate hole states of free atoms. In the method each shell is specified by a subset of orbitals and an occupation number. A suitable choice of starting orbital exponents is found from the equivalent core (EC) approximation. The results of calculations on various hole states of Ne, Na, Mg, and S are in excellent agreement with experimental measurements, and the method should prove to be particularly valuable in the interpretation of Auger and shake-up processes as it is successful for any number of open shells. The use of the one-Hamiltonian and sequential orthogonalization methods in the calculation of particle and hole states has been discussed and a number of SCF methods used in the calculation of excited states are compared in a paper which extends recent restricted Hartree–Fock calculations in high-spin open shells to singly excited states. The concept of localization in molecular excited states is critically examined in an interesting paper which proposes the use of localized molecular orbitals to describe delocalization phenomena. It is demonstrated how such a change from the usual discussion in terms of delocalized molecular orbitals can give a valuable insight into configuration interaction processes.

An algorithm has been suggested to solve the divergence problem of the quadratically convergent SCF method, and the CNDO/2 method with configuration interaction has been successfully applied to the problem of excited-state geometries in a study of the lowest excited singlet state of a number of small molecules. An algorithm for the calculation of electronic structure and spectra using the CNDO/2 CI method has also been published.

The energies of the alternant molecular orbitals, AMO's, for any alternant homonuclear molecule having a singlet ground state have been shown to be related to the conventional Hartree–Fock MO energies by a simple expression involving the correlation correction. A combined AMO and generator–co-ordinate method has been used to study excitations in the electron-gas. Ab initio SCF–LCAO calculations of a number of simple molecules, mainly diatomic species, were found to be sufficiently accurate to give the general character of the electron distribution. A theoretical approach which combines the ideas of the valence-electron and Thomas–Fermi–Dirac theories differs from earlier related treatments in that the core potential is completely non-empirical and is the same for all valence levels. The effects of many-electron rearrangements in excited states on the expressions for the electronic excitation energy have been discussed, and model calculations of transition energies for Be indicate that relaxation corrections are substantial for core excitations, but are small for valence transitions.

The use of Green's functions in the calculation of energy levels from spectral properties is featured in a general review of the application of Green's functions in quantum mechanics. Perturbation treatments based on Green's functions have been used in the calculation of excitation energies in π-electron systems and excited state lifetimes in -atomic systems. The calculation of excited states using the natural transition orbital (NTO) method has been proposed for systems where the ground state is given by a correlated wavefunction. Ab initio calculations of energies. using floating spherical Gaussian orbitals (FSGO's) have been reviewed and their special features well illustrated in calculations for a number of simple molecules using a variety of basis sets of fixed and floating SGO's. The results are compared with those obtained using STO's and GTO's.

A new multiconfiguration method for the calculation of excited states of atoms and molecules has been described. The method is stable and avoids some of the common problems of root flipping and poor convergence by shifting the desired SCI root below the other roots. Test calculations on the 2S states of Li and the 1S states of Be gave results very close to the full CI energies. A different approach is presented by the so-called generalized Brillouin theorem (GBT) multiconfiguration method which is suitable for the calculation of (highly) excited states. Very good agreement with experimental values for excited states of Li is found in test calculations using the method. Test calculations on valence-electron excited states of C, H2O, and CH2O using the GBT–MC method have also been carried out. Optimization procedures in the multiconfiguration SCF method have been discussed.

Projected-unrestricted Hartree–Fock theory has been applied to the calculation of electronic spectra for symmetric molecules. The results of these PUHF calculations for a number of simple molecules using the all-valence-electron (INDO) approximation compare favourably with results obtained using CI methods, and significant improvements are expected from ab initio PUHF calculations.

The effects of electron correlation on the excited-state energies and oscillator strengths of atoms and the methods of calculating them have been reviewed. In the recently developed self-correlated field method, correlation effects are concentrated on electron pairs associated with the same space orbital but with opposite spin functions. When applied to the 2S and 2P ground and excited states respectively of three-electron atomic systems, which· should reveal the effects of the outer electron on the inner pair, it is found that the inner–outer correlation effects are probably more important for the ground state than for the excited state. In neutral and singly ionized first-row atoms there are several states belonging to the 2s2pm configurations which are not the lowest for their symmetry. Calculations of oscillator strengths for transitions involving these states using restricted Hartree–Fock (RHF) methods have not been successful unless charge wavefunctions are used for both the initial and final states. Wavefunctions which do not suffer variational collapse to the lower states of the same symmetry for these non-lowest 2s2pm states have recently been calculated using a non-closed shell many-electron theory (NCMET) and successfully applied to the calculation of oscillator strengths involving O(I) and O(II) transitions, N(I) and N(II) transitions, and C(I) and F(II) transitions. Correlation effects have been reviewed, 31 discussed for the excited states of a ten-electron system, and treated in the spin-density-functional formalism.

A classification scheme for the doubly excited states of a two-electron atom has been given and triply excited states of a three-electron atomic system have been discussed within a formalism whereby many-electron systems are described by a large number of product functions of different configurations.

The calculation of atomic electric dipole oscillator strengths is of considerable current interest and this results, in part, from their relevance to studies in astrophysics and plasma physics. There have been a number of developments in this area, including the formulation of a reasonably complete non-relativistic many-body theory which gives accurate results for single and multiple excitations to discrete and autoionizing states in atoms. The first-order theory of oscillator strengths (FOTOS) is able to predict the electron correlations which are important to the oscillator strength and has been successfully applied to one-and two-electron excitations in Li(I), N(I), and N(II) and to a number of other systems. An important step in the theory is the derivation of 'first-order symmetries' by finding out the type of angular distortions which are allowed for initial and final states owing to the symmetry of the dipole operator. Taking the authors' example, the first-order symmetry distortion of the Be ground state whose Fermi-sea (a set of spin orbitals configurations are 1s22s2, 1s22p2 yields for the 1P0 excited states the following two symmetry 'configurations': (sssp), (sspd). For the Be 1s22s2p1Po excited state the corresponding first-order symmetry configurations are: (sspp), (ssss), (sssd). The third one of these is discarded because it does not yield an overall 1S symmetry. The calculation proceeds by calculating the radial function for each of the allowed symmetries. The theory also has semiquantitative value in that it can predict those electron correlations which are important in photoelectron spectroscopy, in the photoionization of noble gases, and on molecular transition probabilities.

Many-body perturbation approaches to the calculation of transition probabilities have been reviewed. The oscillator strengths of a number of doubly ionized elements of astrophysical interest have been calculated as have those for the 2s3p-2s3d transition in C, N, and O using a model Hamiltonian which includes an effective potential for an optical electron in a many-electron atom and in a positive ion. The calculation of oscillator strengths using a model potential method has been compared with the quantum defect method. The important and much debated question of which of the three formally equivalent expressions for the non-relativistic oscillator strengths, written in the length, velocity, or acceleration forms respectively, is the more appropriate has been clarified in a paper which identifies conditions for which good agreement is found between all three expressions. This is illustrated by calculations on the 1s22s22p2Po- 1s22s2p22D transition in C(II) and O(IV).

Atomic Systems. — Of the numerous papers concerned with the calculation of excited-state energies and oscillator strengths in particular atomic systems, only a few are reported on here. A generalized oscillator strength equation has been applied to the case of initially excited H-like atoms. The FOTOS method was used to calculate the oscillator strength of the Li 1s22s 2S-2Po transition in a paper which is concerned with highly excited states of atoms which interact with Rydberg and continuum states of the same symmetry. A comparison of non-relativistic and relativistic oscillator strengths for transitions in singly ionized Si, Ge, Sn, and Pb shows the importance of spin-orbit interactions of the optical electron. The Green's function method has been used to calculate mean excitation energies of ground- and excited-state hydrogen-like atoms and a number of papers deal with aspects of the excited states of helium, namely the behaviour of interelectronic angular distribution functions, CI wavefunctions for the 3s state, solutions to the Schrodinger equation, and model calculations relevant to studies of non-adiabatic core polarizations and penetration corrections in alkali-like atoms. A technique to calculate approximate Brueckner or natural orbitals for systems with a single valence electron has been used to calculate the hyperfine interaction of some excited states of alkali atoms, and non-relativistic Hartree-Fock solutions for excited states of In+ in the form of analytical expansions in STO's have been obtained.

Molecular Systems.-A many-body perturbation theory suitable for the calculation of molecular excited states has been used to calculate the two lowest singlet and triplet states of Σg+, Σu+ Πg, and Πu symmetries for molecular hydrogen with very good accuracy. The method uses Brandow expansion diagrams, which can be used for degenerate states and which are a development from the linked Goldstone perturbation diagrams used in calculations of energy levels in atomic systems. SCF-CI calculations of the B"Σu+ and D'Πu states of the hydrogen molecule were carried out to give the values of the electronic wavefunctions at the crossing point of their potential curves. The values were required in a study of collision-induced predissociation. Calculations on the uncharacterized third 1Σg+ state of H2 show that its Born-Oppenheimer potential curve has a double minimum. The inner (Rydberg-like) minimum occurs at about 1.99 bohr and the deeper outer ('ionic') minimum at 3.30 bohr. The state has been assigned the spectroscopic term symbol [MATHEMATICAL EXPRESSION OMITTED]: this accounts for spectral features previously assigned to separate G and K electronic states. Non-exponential wavefunctions in elliptical co-ordinates for a number of electronic states of the molecular hydrogen positive ion, H2+, have been described. A calculation of the 2Σu+ first excited state of H2+ has been made using a modified form of the MS-MA perturbation theory which improves its short-range performance. An excimer-like state has been identified in a study of the ground and low-lying excited states of H4 formed by the rectangular approach of two H2 molecules. Ab initio SCF-MO-CI calculations were carried out and one of the excited states of H4 could be correlated with ground (X'Σg+) and first excited singlet (B'Σu+) states of the H2 molecule. The results are illustrated in Figure 1.