Electronic Calculations on Large Molecules

Radom and J. A. Pople, in Theoretical Chemistry , ed. W. Byers Brown, MTP International Review of Science, Physical Chemistry Series One, Volume 1, Butterworths, London, 1972, p. 71. 

5 Computational Methods for Large Molecules and Localized States in Solids , ed. F. Herman, A. D. McLean, and R. K. Nesbet, Plenum Press, New York, 1973. 

6 Wave Mechanics - the First Fifty Years , ed. W. C. Price, S. S. Chissick, and T. Ravensdale, Butterworths, London, 1973. 

These developments will be the main subject of this Report. 

Introduction.—The LCAO-MO method remains the most important approach for evaluating wavefunctions for large molecules in spite of its known defects. The details of the method are well known. The main outlines are given here only to establish the nomenclature. The method attempts to describe the wavefunction as a single Slater determinant comprised of one-electron space-spin functions or spin orbitals  

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Electronic Calculations on Large Molecules Closed shell 2n electrons... 

The carbon chemical shifts for steroids are the most readily available data from a routine 13C NMR determination. Since they reflect the electronic and steric environments of the various carbon nuclei, they provide sensitive insights to the configurational and conformational features of such molecules. While much interesting work on ab initio molecular orbital calculations of carbon chemical shifts is now appearing, it is probably true that the difficulties of carrying out such calculations on large molecules will prevent their applications to steroids for some time. We are limited, therefore, to a more empirical approach to steroid carbon chemical shifts. (3, 38)... 

Another method which takes into account electron correlation is density functional theory (DFT), which is implemented in Gaussian with the label B3LYP. The basic idea of DFT is to calculate the energy of an atom or molecule in terms of electron density rather than the wavefunction tf/ The method has become increasingly popular because it can do reasonably accurate calculations on large molecules in significantly less time than that for the HF method. 

An entirely different way to treat the electronic structure of molecules is provided by valence bond theory, which was developed at about the same time as the molecular orbital approach. However, valence bond theory was not so amenable to calculations on large molecules, and molecular orbital theory came to dominate electronic structure theory for such systems. Nevertheless, valence bond theories are often considered to be more appropriate for certain types of problem than molecular orbital theory, especially when dealing with processes that involve bonds being broken and/or formed. Recall from Figure 3.2 that a self-consistent field wavefunction gives a wholly inaccurate picture for the dissociation of H2 by contrast, the correct dissociation behaviour is naturally built into valence bond theories. 

The calculation of NMR chemical shifts at the uncorrelated HF-SCF level is nowadays routinely possible using CPHF theory. Efficient implementations have been reported in particular within the GIAO approach [35,91] and (using integral-direct techniques) calculations on large molecules consisting of more than 100 atoms are routinely possible [91]. Recent examples include HF-SCF calculations on hexabenzocoronene derivatives [24,25] as well on aluminum halide clusters [18]. The status here has been earlier reviewed by others and thus will not be repeated. Our focus will be instead on electron-correlated approaches, in particular, as consideration of electron-correlation effects is essential in many cases. 

The most promising approaches for efficient electronic structure calculations on large molecules are generally based on density functional theory with Kohn-Sham orbitals [32-35]. The most efficient such method for CE-BEs is based on Koopmans theorem, but this approach has quite limited accuracy [36-39]. Better accuracy can be obtained from calculations based on an effective core potential [40-45], an equivalent core approximation [46-48], a fractionally occupied transition state [49-52], or with a ASCF approach [29, 31, 53-57]. Time-dependent density functional theory is also widely used for CEBE calculation [58-62], wherein the best results are usually given with functionals having a long-range correction [63, 64]. 

J. Cioslowski, Ab Initio Calculations on Large Molecules Methodology and Applications, Reviews in Computational Chemistry, Vol. 4, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley, New York, 1993, p. 1. Also see J. Cioslowski, Electronic Structure Calculations on FuUerenes and Their Derivatives, Oxford University Press, Oxford, 1995. 

A method to speed up MP2 calculations on large molecules is the local MP2 (LMP2) method of Saebp and Pulay [S. Saebd and P. Pulay, Annu. Rev. Phys. Chem., 44, 213 (1993)]. Here, instead of using canonical SCE MOs in the Hartree-Fock reference determinant d>o. one transforms to localized MOs (Section 15.8). Also, instead of using the virtual orbitals found in the SCF calculation as the orbitals a and b in (16.13) to which electrons are excited, one uses atomic orbitals that are orthogonal to the localized occupied MOs. Also, in (16.13), one includes only unoccupied orbitals a and b that are in the neighborhood of the localized MOs i and j. 

As noted in Section 15.16, calculation of 2-electron repulsion integrals in SCF MO calculations on large molecules can be speeded up by expanding products like, (1 ) j(l)... 

However, since this expression involves the transformation of the two-electron integrals to the MO basis, it is not well suited to calculations on large molecules. A more useful expression is obtained by introducing in (10.7.1) the one-electron AO density matrix... 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

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