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Photochemistry Volume 11
A Review of the Literature published between July 1978 and June 1979
By D. Bryce-Smith The Royal Society of Chemistry
Copyright © 1981 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84755-462-8
CHAPTER 1
Part I
PHYSICAL ASPECTS OF PHOTOCHEMISTRY
1
Spectroscopic and Theoretical Aspects
BY R. DEVONSHIRE
1 Introduction
The format of this report is similar to that of my last contribution. In general, emphasis is placed on intramolecular photophysical processes. Several aspects of photochemical excited-state decay processes are covered in later chapters of this volume.
In the two-year period covered in this report there have been a number of significant developments, amongst which are the following: (i) the relentless progress of ab initio methods of calculating excited-state properties as exemplified by calculations on a model of the active site in the iron-sulphur protein rubredoxin, (ii) fresh impetus and progress towards methods of calculation which avoid the Bom-Oppenheimer separation, (iii) a major study of cyclic π-electron systems that uses magnetic circular dichroism, which should advance the wider application of this technique, (iv) the development of the concept of a proximity effect to interpret the photophysical properties of N-heterocyclics, (v) the remarkable observations of mode-to-mode vibrational-energy flow in the first excited singlet states of benzene and aniline, and (vi) the probing of homogeneous linewidths in fluid solutions in studies of the anti-Kasha fluorescence of aromatic hydrocarbons.
2 Calculations of Electronically Excited States
Methods. — The application of ab initio quantum-mechanical calculations to problems of interest to experimental photochemists has been discussed briefly. The effect of the choice of basis set, including the addition to the basis of so-called 'polarization functions', and of the extent of configuration interaction on calculations of electronic spectra, the effect of substituents and geometry upon the relative energies of states, and of potential surfaces for chemical reactions of excited states are all illustrated by examples taken from the recent literature. A new restricted self-consistent field (RSCF) method has been proposed for the calculation of excited states. The excited-state wavefunction obtained from RSCF guarantees the orthogonality with the Hartree–Fock (HF) ground state and satisfies a number of Brillouin-type theorems. Cluster expansion is a technique for obtaining exact wavefunctions from approximate ones. When applied to open-shell systems it takes the form of the symmetry-adapted cluster (SAC) expansion method. The ground state ψg of a given spin-space symmetry is given in equation (1), where ηg is the normalization factor, θ the symmetry projector, CI, the
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
coefficients obtained from variational equations, SI+, the symmetry-adapted excitation operator, and Θ0 a reference wavefunction. It has been shown that the set of functions {ΘK}, where Θ0 = ηkPΘSk+ψ'g (P = 1 - |ψg x ψg|), forms a basis for the excited states of the same symmetry as the ψg and can lead to exact excited states. The method can be extended to excited states having symmetries different from the ground-state symmetry. A new method for calculating SCF wavefunctions for the excited states of atoms and molecules has been proposed in which for every iteration a number of diagonalizations of matrices representative of different operators are undertaken. The procedure results in a rapid convergence to the correct SCF limit. The variational bounds to excited states calculated from a generalized Brillouin-Wigner equation and from a projection operator method have been investigated.
A much improved scheme for the calculation of multiconfiguration self-consistent field (MCSCF) wavefunctions has been published recently. The authors assert that their scheme is a superior one for constructing computer programs for the MCSCF method and that it leads to programs with maximum efficiency. The authors introduce an exponential transformation for the trial state functions in a manner similar to an exponential spin–orbital transformation previously used in the method and which is intended to improve the convergence properties.
A modified Rydberg formula for atomic states has been suggested that gives accurate energies of [core] (ns21S) terms for a wide variety of atoms. The method can also be applied to [core] (nlnl' 2s+1L)] terms and extended to include configurations of the type [core] npN (N = 1 — 6). In HF calculations the core orbitals are frequently replaced by an effective potential or pseudopotential. An analysis of the use of pseudopotentials in local-density (LD) formalism calculations shows that their application there may be more appropriate than in HF calculations where more approximations are involved. An LD pseudopotential treatment of several excited and ionic states of first-row atoms showed errors due to the frozen core to be less than 10-3 hartree. The first-order HF equations of the 1s2p3s4P0 state of three-electron atomic systems have been solved exactly using the nuclear charge expansion method. There is only limited experimental information available at the present time on these doubly excited states of the lithium isoelectronic sequence, but discrepancies have been noted between values calculated previously using a variation method and experimental assignments. The discrepancies may arise from interactions between the near-degenerate states 1s2s3p4P0, 1s2p3s4P0, and 1s2p3d 4Po(vide infra). The problem of variational collapse in calculations of excited states which are not the lowest of their symmetry is central to a debate in the recent literature. In response to a paper critical of their earlier work two authors have argued that their method for truncated Hamiltonian matrices, which chooses the root that minimizes the energy as well as the correlation overlap , yields upper bounds and does not suffer variational collapse. Their polemic has prompted a reply. The Galerkin-Petrov (GP) method involves the application of the method of moments to obtaining approximate solutions to operator equations such as the Schrodinger equation. It has been shown that with care the GP method can be used successfully to calculate excited states of the same symmetry classifications as the ground state.
A non-relativistic SCF method for calculating the higher excited states of atoms has been developed, and electron correlation in the ground and excited states of first-row atoms has been discussed. The extension of the Madelung rules for the order of the filling of successive electron shells in the ground states of neutral atoms to excited states has been investigated. The two general rules state that (a) the electron shells are filled in the order of increasing values of the quantum number sum n + l, where n is the principal quantum number and l the azimuthal quantum number of the shell (nl), and (b) for electron shells with the same value of n + l ([equivalent to] k in the notation of the author) the shells are filled in the order of increasing n or decreasing l values. These rules were found to hold exceptionally well for the levels of the valence electron in the alkali-metal atoms and for singly ionized alkaline-earth atoms. It is suggested that there may be a group theoretical explanation for this k ordering. The author extended his observations to include a large number of other 'one-electron' spectra and confirmed the validity of k as an 'energy-ordering' quantum number and of the characteristic patterns for groups with constant k. Figure 1 shows the regions of validity of k ordering and hydrogenic (H) ordering.
The calculation of the excited states of closed-shell systems has been investigated using the direct minimization SCF technique, which is based on the two-by-two rotation of orbitals. The technique was applied to the excited states of H2O and FNO and showed, surprisingly, that the relaxation of doubly occupied orbitals, in contrast to that of the excited-state orbital, always gives a significant contribution to the lowering of the excited-state energy in the SCF process. This observation calls into doubt electron-hole potential (EHP) calculations in which the relaxation of ground-state orbitals via mixing with the virtual ones is disregarded. The EHP method has recently been applied to problems of photochemical interest, using various standard parameter sets (CNDO/S, INDO/S, MINDO/2), and in spite of the observations in the paper above the results were regarded as satisfactory. A form of the Rayleigh-Schrödinger perturbation theory has been developed which can be applied to the calculation of low-lying excited states. In contrast to the many-body perturbation theory (MBPT), where Slater determinants are used as the unperturbed states, it is able to treat corrections to a multiconfiguration function of zero order.
The local approach for the calculation of electronic correlation energies in molecules has been generalized to excited states. The method was applied to a system for which an exact solution was available (H6 ring) and found to give results as good as those expected for ground-state calculations. The application of semi-empirical LCAO–SCF–MO calculations to π-electron systems is known to have a number of shortcomings. These probably arise from differences in the Σ-Π interactions in singlet and triplet states and from electron correlation effects. The Σ-Π interactions have been included explicitly in a derivation of the effective Π-electron Hamiltonian for excited states of π-electron systems. In calculations of π-π* excitation energies using this Hamiltonian the PPP method is used for the ground state. Then the effective electron-repulsion integrals are evaluated, and finally a singly excited configuration interaction method (SECI) is used, in which σ-σ* and π-pi]* configurations interact through exchange-type MO integrals over π- and σ-MOs. In a later paper the authors consider the effect of the ΣΠ interaction on the oscillator strengths of π-π* transitions in benzene and naphthalene derivatives. An all-valence LCAO-MO-SCF-INDO method has been applied successfully to the calculation of electronic transition energies and ither molecular properties, and a recent observation that equations (2) and (3) (where Emolecule is the total energy of the molecule and Vne and Vnn are the nuclear-electronic and nuclear-nuclear interaction energies, respectively) accurately (~1%) hold for the ground states of a large group of neutral molecules has been extended to consider excited and ionic states, where they were found to be just as accurate approximations.
Results. — No attempt has been made to report on the extensive literature on atomic systems, and this has made it possible to increase coverage of the results of calculations on molecular systems.
An interesting feature of the electronic spectroscopy of molecular hydrogen is that the lowest 1Σg+ excited state is characterized by a double-minimum internuclear potential. Accurate calculations of this potential have previously given satisfactory agreement with the experimental vibrational-rotational spectra of this E,F1Σg+ state. Following the original observation of this double-minimum potential it was predicted that the next three 1Σg+ excited states would behave similarly. A recent calculation of the Born-Oppenheimer potential of the second excited 1Σg+ state, designated G,K1Σg+, has found such minima, and the potential has subsequently been used to calculate its vibrational levels. There is poor agreement with the experimentally observed vibrations of the G and K states, indicating the necessity of adiabatic corrections to the potential. These double minima can also be seen in the results of ab initio calculations of the potential curves and wave functions of a number of excited states of H2. The calculations are based on the demi-H2+ model for Rydberg states and include a complete configuration interaction (CI) calculation. The double minima are discussed in terms of the occupation numbers of the natural spin orbitals. Interestingly, the curves of the occupation numbers with respect to internuclear distance display a non-crossing rule in analogy to that for potential curves. The van der Waals region (5 — 10 a0 of the lowest excited state of H2, 3Σu+, has been investigated using the modulated form of the second-order Murrell–Shaw–Musher–Amos (MS-MA) perturbation theory. It is concluded that the 'physical' interactions are more difficult to calculate than those determining the chemical bond. Ab initio calculations, using a small basis, have been used to investigate the Rydberg and valence-shell characters of the low-lying B1Σu+ and b31Σg+ states of the H molecule. A recent paper has reported the measurement of an electronically excited state of H3. The observed levels were attributed to vibrational levels in the upper Jahn-Teller sheet of H3, and their energies and intensities were compared with existing calculations.
The Born–Oppenheimer potential surfaces for the three lowest singlet electronic states of the H4 molecule with trapezoidal geometry and for the four lowest singlet states of the H4 molecule with 'triply right' (C2v) tetrahedral geometry have been calculated from an ab initio valence-bond (VB) method employing a minimum basis set (STO-4G). The particular interest of the results is their relevance not only to aspects of the photochemistry of H2 and H4 and the general problem of useful representations of six-dimensional space but also to the modelling of more complex systems where analogous topology may be identified. To illustrate the latter point the authors draw analogies between trapezoidal H4 and a reaction in organic photochemistry effectively between four electrons in four atomic orbitals referred to as a 2s + 2s pericyclic reaction, and in the case of tetrahedral H4 geometry they draw attention to the propensity for diagonal bonding in the S1 state of a pericyclic array of 4N interacting atomic orbitals. This interesting theme is continued in another paper that models organic photochemical reactions on H4 excimer states.
The spectroscopy of the rare-gas dimers is clearly of considerable current interest to research into ultraviolet lasers. A timely paper has presented ab initio CI calculations of the potential curves of sixteen of the lowest excimer states of Ar2. The results agree well with the limited experimental information available for these states. Of similar interest are SCF calculations of the ground 1Σg+ state of Xe2, the first four states of the Xe2+ ions, and eight excimer states of Xe2. The calculations use an approach to heavy-atom calculations, in which the core electrons are replaced with an effective potential that also includes relativistic effects. The application of perturbation theory to the study of the electronic structure of rare-gas halide excimers has been shown to give quantitatively similar results to previous ab initio calculations. The electrostatic model used in the paper includes mixing between the ionic excimer and neutral ground configurations, which proves to be significant to the crossing of the excited Σ and Π curves. Of the rare-gas monohalides only the neon monohalides have not been studied experimentally owing to the short wavelength of the fluorescence. Following calculations of the potential-energy curves for the lowest four states of NeF and of transition moments, the fluorescence-emission spectrum of NeF has been predicted. It is dominated by the charge-transfer band 21Σ [right arrow] 12Σ at 108 nm, and the result is shown in Figure 2. Further information of interest in the design of laser systems lies in the results of ab initio calculations of the potential-energy curves of Ne2F. There is both experimental and theoretical evidence for the existence of a state in the analogous molecule Ar2F, and it is reasonable to expect the ionic states of the Ne2F molecule to be even more strongly bound. Calculations using a valence double-zeta basis set and an adequate CI showed that indeed the Ne+ F-2 state was bound by 0.76 eV relative to its lowest dissociation limit, Ne + 22Σ+NeF. Ab initio electronic potential-energy surfaces for doublet HeH and comparisons between the theory and experimental observations of the visible excimer bands of potassium- and sodium-noble-gas molecules have been published.
(Continues...)
Excerpted from Photochemistry Volume 11 by D. Bryce-Smith. Copyright © 1981 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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