Chapter 1 Ab initio Calculations on Molecules containing Five or Six Atoms By c. Thomson, 1, 
Chapter 2 Theories of Organic Reactions By A. J. Stone, 39, 
Chapter 3 The Quantum Mechanical Calculation of Electric and Magnetic Properties By A. Hinchliffe and D. G. Bounds, 70, 
Chapter 4 The Use of Pseudopotentials in Molecular Calculations By R. N. Dixon and I. L. Robertson, 100, 
Author Index, 135,

CHAPTER 1

Ab initio Calculations on Molecules containing Five or Six Atoms

BY C. THOMSON

1 Introduction

The present Report attempts to survey calculations carried out during the past few years by ab initio methods which were not covered in Volume 2 of the present series. In the latter, the Report by Thomson dealt with molecules containing up to four atoms, and the article by Duke dealt with large molecules. Since the completion of Volume 2, the literature on ab initio calculations has continued to grow rapidly and it was soon clear that space limitations would preclude any comprehensive coverage of the literature dealing with medium-sized molecules. Therefore this Report is restricted to calculations on molecules containing five or six atoms, and even within this group it is not possible to refer to all such calculations which have been published. Those studies which seem of particular interest to the Reporter have therefore been surveyed so that the selection is somewhat subjective.

During the past five years there have been no spectacular advances in fundamental theory; rather there has been a consolidation of earlier experience in ab initio methodology and a more widespread use of existing methods in tackling problems of interest to more chemists in general, i.e. there have been many more applications to medium sized polyatomic molecules, usually employing minimal basis sets (MBS).

There have also been many more calculations in which geometry optimizations are carried out, and in which the basis sets in SCF calculations have been extended to DZ or DZ+ P quality, and more recently one sees an increasing use of methods which include at least some electron correlation, especially via configuration interaction (CI). Examples of the latter calculation were until recently restricted to molecules containing up to three atoms, but the recent development of efficient CI programmes had enabled these calculations to be carried out without too great expense on a variety of larger molecules, and this work is referred to later on in this Report.

In order that this Report be useful to non-specialists interested in earlier ab initio work in this area, it is useful to cite several reviews and books relevant to the subject matter of this chapter. The bibliography by Richards and co-workers has been updated to 1973, and contains a list of all the earlier ab initio calculations. The proceedings of the First International Congress on Quantum Chemistry, and that of a conference on 'Quantum Chemistry: The State of the Art', contain many review papers and survey many of the currently interesting areas in quantum chemistry. A volume devoted to theoretical chemistry has appeared in Series Two of the MTP International Review of Science, and an excellent survey of recent developments in molecular electronic structure theory by Schaefer has recently appeared. This review gives a more comprehensive list of books and reviews than is possible here. We should, however, mention that a comprehensive series of eight volumes on 'Modern Theoretical Chemistry' is starting to appear and this series in particular should give an up to date and comprehensive survey of ab initio calculations. We have also not attempted in this Report to survey the individual molecules containing five and six atoms which are studied usually together with the larger molecules in the series of papers from Pople's group. Recent reviews of this work have appeared and the reader is referred to these for further details and references. The general availability of the Gaussian 70 programme developed by Pople and co-workers (via the Quantum Chemistry Program Exchange has encouraged many non-specialists to venture into this field and to extend their investigations to larger molecules. However, it is important that such packages are not used in an uncritical way, and the limitations of the SCF procedure, and of minimal basis set calculations in certain instances, should be borne in mind. A recent book by Csizmadia is useful in this light, dealing with applications to organic molecules.

As in the previous Report, developments in theoretical and computational methods as such will not be dealt with. The results of calculations will usually be quoted in atomic units (distances/Bohr, energies[Hartree) but occasionally electron volts (eV) or kilojoules (kJ) for energies are used. A list of commonly used abbreviations is given at the beginning of this volume.

The calculations described are organized into sections defined by the general formulae of the species. This is to some extent an arbitrary division but serves to group together those molecules of similar geometrical structure. As mentioned above, discussion will be restricted usually to work carried out during the period 1973–6.

2 Molecules containing Five Atoms

These are divided into the following classes, where in a particular class we also consider the relevant charged species: H5, AH4, AB4, HAB3, H2NX, H2CNX, Nitrenes, Diazomethane, H2CXY, Carbonium ions, Miscellaneous penta-atomic molecules.

A. H5, H5+, and H5-. — The simplest penta-atomic molecule is H5+ and it has been the subject of several recent studies. The mass spectrum is well known and earlier work on the stability of this molecule is referred to in a paper by Huang et al.

These authors investigated several geometrical structures by either carrying out a VB calculation with CI, or by obtaining SCF wave functions using a flexible basis set. The VB-CI calculations showed no stability for H5+ in a D2a configuration (in contrast to previous predictions by Poshusta et al. and the authors concluded that the method is unreliable for this type of ionic system. However, the SCF calculations predict a binding energy of 0.007 Hartree (17.8 kJ mol-1) with an overall C2v symmetry. A more recent VB-CI study by Salmon and Poshusta used a more flexible basis set and gave similar results to the SCF calculations, and it is clear that polarization of the basis orbitals is very important in improving the VB results. Other calculations on H5+ and Hn+ (n < 15) have been reported, but the most extensive work to date is that of Ahlrichs, who used the PNO-CI and CEPA methods. Reviews of these methods have been given elsewhere, but essentially they go beyond the SCF–type wave function and include electron correlation. In the PNO-CI method, all doubly excited configurations in addition to the HF function are included, and the CEPA method also accounts for the effects of higher than doubly substituted configurations in an approximate way. For H5+ the two methods give very similar results.

In Ahlrichs' work, a large CGTO basis of lobe functions was used and the orbital exponents were carefully optimized so that the various different kinds of interaction such as ion-dipole, ion-quadrupole, dispersion, etc. were all accounted for. It was claimed that the relative energy errors should be no larger than 10-3 Hartree. Various geometrical configurations were investigated and, for the most important of these, geometry optimizations were carried out.

The minimum energy structure was found to be of D2d symmetry whereas the HF geometry is of C2v symmetry, as found in other work. The potential surface near the D2d structure is, however, extremely shallow. The author concludes that at room temperature the structure is mainly H2H3+. The computed value of De is 0.012 Hartree (30.9 kJ mol-1) and [FORMULA NOT REPRODUCIBLE IN ASCII].

B. AH4. — CH4and CH4+. The number of calculations on CH, listed in Richards' bibliography is 75 (up to 1973), and most of the current methods in use in quantum chemistry have been tested on this molecule.

Several calculations on CH4, have been concerned with the calculation of innershell or outer-shell ionization energies.

The most reliable method has proved to be the ΔSCF method, in which the core binding energies are obtained by subtracting the SCP energies of the ground state from the SCF energy of the system with one of the core electrons removed.

Several papers have dealt with the calculation of the K-shell ionization energies of CH4, and also the Auger spectrum. Bagus and co-workers have reported two such studies. In the first paper, 25 the authors computed a 1A1 ground-state energy of -40.207 34 Hartree, using a (10,6/6,1) [right arrow] [7,5/5,1] basis set supplemented by Rydberg functions on both C and H. Calculations of the Rydberg states of the 1a1-1 hole state were carried out and the results were in good agreement with experiment.

In the second paper 26 a more extensive basis set (12,7,2/6,2) [right arrow] [8,5,2/5,2] gave an SCF energy of -40.214 178 Hartree for R(CH) = 2.066 Bohr. The computed Auger energies were in excellent agreement with experiment and the authors conclude that SCF wave functions provide a qualitative basis for the analysis of molecular Auger spectra. It is emphasized that multiplet splittings of the final states are important, and play a key role in determining the spectra.

Deutsch and Curtiss have investigated in more detail how the core ionization energies of CH4, H2O, NH3, and HF depend on the size and completeness of the basis set. Calculations were carried out both with the RHF and UHF procedures and eight different basis sets from a minimal to essentially a DZ + P basis set. The authors conclude that a large and flexible basis set is needed to obtain good agreement with experiment, with polarization functions being less important for highly symmetrical molecules like CH4.

A related topic is the computation of valence-shell ionization potentials (VSIP). The calculation of vertical ionization potentials via Koopmans' theorem leads in many cases to serious errors, and a version of the ΔSCF method has been used to compute VSIP for several small molecules, including CH4. All the valence hole states of the molecule were computed. Agreement with experiment was substantially better than in the calculations using Koopmans' theorem.

Other authors have used less accurate SCP wave functions to investigate various other properties of CH4 such as the force constants, previously studied by Meyer and Pulay, using Gaussian lobe functions. Schlegel et al. have used the popular STO-3G and STO 4-31G basis sets to compute force constants in a variety of first- and second-row hydrides, including CH4, and SiH4. The 4-31G basis set gives reliable values for the harmonic constants but the STO-3G basis does not. The higher force constants were also investigated. A more recent paper has used the same programme and a 4-31G basis for studies on several hydrocarbons. Similar results were obtained, but the bond lengths and angles were more extensively optimized in order to compute the equilibrium geometries.

The one-electron properties such as the deuteron quadrupole coupling constant have been less studied until recently, but Dixon et al. have recently reported the results for this property and also values of the diamagnetic shielding and susceptibility computed with a medium-size basis set. Moderate agreement with experiment was obtained.

Several papers have dealt with the evaluation of wave functions including correlation in various ways. Birnstock has calculated the 13C shielding constants in CH4 and several other small molecules using an approximate form of uncoupled Hartree–Fock theory and the minimal basis set wave functions of Palke and Lipscomb. The results were similar to those obtained earlier by Ditchfield et al.

CI calculations have also been reported of the states involved in Auger transitions in CH4. Using a (9,5/5) [right arrow] [5,3/3] basis set and valence-shell CI, good agreement was obtained with experiment, and also with the earlier SCF calculations. The authors also used the same method for the Auger spectra of HF, H2O, and CO and it is clear that a modest CI using a medium size basis is capable of describing these spectra.

There has been renewed interest in recent years in the calculation of Compton profiles and momentum expectation values. Much of the earlier work involved the use of SCF wave functions, but the results obtained using large-scale CI wave functions have recently been published. It was concluded, however, that it is not necessary to go beyond the near-HF wave function in order to compute reasonable profiles, providing large, well balanced basis sets are used, a conclusion also reached by Tanner and Epstein.

The problems involved in the calculation of nuclear spin-spin coupling constants are well known, and Roos and co-workers earlier used a perturbation procedure to calculate JHH or H2 with encouraging results using ab initio wave functions. The method involves correlating the zeroth-order wave function by a large CI calculation and treating the coupling between protons by perturbation theory involving the excited triplet configurations. Since this approach should be applicable also to polyatomic molecules, the authors 41 have studied CH4, H2O, and NH3, using GTO basis functions. Several basis sets of different sizes were used. It is clear that correlation effects play an important role in the case of indirect coupling between nuclear spins separated by two bonds, and the basis set is also very important. Except for H2O, however, the agreement with experiment was not very good, and it was suggested that vibrational effects might be significant. It is crucial to include the doubly excited triplets in the calculations.

The calculation of a substantial fraction of the correlation energy, particularly the valence-shell correlation, and the detailed analysis of the contributions to the correlation energy now seems to be feasible for small polyatomic molecules and Ahlrichs and co-workers have reported detailed results on BeH2, BH, BH3, CH3-, CH4, NH3, H2O, and OH3+. Discussion here is confined to the results for CH4. The calculations were by the PNO-CI and CEPA-NO methods, details of which were given in ref. 21. The basis sets used were ca.DZ + P or a larger basis with both d and f functions, but two sets of p functions on H. These basis sets were bigger than in previous calculations by this method, and the authors claim to have obtained ca. 85% of the correlation energy in each case. For CH4, the best energy obtained was -40.425 22 Hartree.

Ahlrichs has also described in more detail the method used in this work. Werner and Meyer have continued work with their version of the PNO-CI and CEPA methods, computing the static dipole polarizabilities of CH4. Various basis sets were used in this study, mostly with better results than in earlier work. Meyer also studied the energy surface of CH4+ in earlier work with this method.

MC-SCF calculations on polyatomic molecules are still rather rare, although there have been many such calculations on triatomic and diatomic molecules. Levy has described the results of such calculations using a minimal STO basis set for CH4, C2H4, and C2H6. A quadratically convergent method was described and the results of localizing the orbitals were investigated.

The GVB method developed by Goddard and co-workers has been applied to alkanes, ethylene, and acetylene, and CH4 was among the molecules studied. The advantages of this type of wave function were discussed in Volume 2, and in the current work minimal, DZ, and DZ + P basis sets were used. One interesting observation is that heats of reaction for reactions involving the breaking of single bonds are quite reasonably described, i.e. for the reaction (1) ΔHSCF = 363.6 kJ mol-1 for

DZ + P basis, whereas the GVB value is 409.2 kJ mol-1 the experimental value being 430.9 kJ mol-1 One feature of interest for CH4 is that the hybridization is sp2.1, rather than sp3 as in the usual VB description.

Potential energy (PE) surface calculations are now becoming feasible, though expensive, for larger systems and several authors have described work on CH4 in this connection. Eaker and Parr have used the diatomics in molecules method (DIM) to obtain the potential energy surfaces for CHn and obtained a heat of atomization which was ca. 84 kJ smaller than experiment. Wiberg and co-workers have studied the energy changes for four angular deformation modes of CH4 using STO-3G, STO 4-31G, and DZ + P basis sets. The 4-31G basis set appears to be capable of giving a reliable description of the bending modes and these were related to those involved in the formation of several types of small-ring compound.

(Continues...)

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Theoretical Chemistry

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Theoretical Chemistry Volume 3

A Review of the Literature Published up to the End of 1976

The Royal Society of Chemistry

Contents