Wavefunction-based Methods

1 Wavefunction-based Methods. - Within the Hartree-Fock approximation, the solution to Eq. (2) is approximated as a single Slater determinant, 

Within the Hartree-Fock approximation, the total electron density becomes 

Solving the Hartree-Fock equations (5) yields in most cases more than N orbitals and one may accordingly construct improved approximate (so-called Cl, configuration-interaction) wavefunctions 

are the so-called natural orbitals, and their occupancies , lie in the interval [0 1]. Per definition, the improvements that are obtained when replacing Eq. (4) by Eq. (11) are the correlation effects. These may be included either through application of the variational principle or perturbatively. 

In order to solve the Hartree-Fock equations, one usually expands the solutions in some finite set of basis functions, 

---

The chemistry of superheavy elements has made some considerable progress in the last decade [457]. As the recently synthesized elements with nuclear charge 112 (eka-Hg), 114 (eka-Pb) and 118 (eka-Rn) are predicted to be chemically quite inert [458], experiments on these elements focus on adsorption studies on metal surfaces like gold [459]. DFT calculations predict that the equilibrium adsorption temperature for element 112 is predicted 100 °C below that of Hg, and the reactivity of element 112 is expected to be somewhere between those of Hg and Rn [460, 461]. This is somewhat in contradiction to recent experiments [459], and DFT may not be able to simulate accurately the physisorption of element 112 on gold. More accurate wavefunction based methods are needed to clarify this situation. Similar experiments are planned for element 114. 

One specific problem becomes very acute in wavefunction based methods the basis set problem. The introduction of a finite basis set is not highly problematic in HE theory since the results converge quickly to the basis set limit. This is, unfortunately, not true in post-HE theory where the results converge very slowly with basis set size - which is another reason why the methods become computationally intractable for more than a few heavy atoms (heavy being defined as nonhydrogen in this context). These problems are now understood and appropriate approaches have been defined to overcome the basis set problem but a detailed description is not appropriate here. 

Table 6.15 Comparison of experimental barriers at 0 K (kJ/mol) for the addition of methyl radical to alkenes CH2=CXY with those calculated with wavefunction-based methods.

The mathematical term functional, which is akin to function, is explained in Section 7.2.3.1. To the chemist, the main advantage of DFT is that in about the same time needed for an HF calculation one can often obtain results of about the same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical applications of DFT are but one aspect of an ambitious project to recast conventional quantum mechanics, i.e. wave mechanics, in a form in which the electron density, and only the electron density, plays the key role [5]. It is noteworthy that the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3), largely for his role in developing practical wavefunction-based methods, and Walter Kohn,1 for the development of density functional methods [6]. The wave-function is the quantum mechanical analogue of the analytically intractable multibody problem (n-body problem) in astronomy [7], and indeed electron-electron interaction, electron correlation, is at the heart of the major problems encountered in... 

Current post-Hartee-Fock OEP methods were shown to handle successfully such cases for which other strategies to approximate the exchange-correlation potential fail.82 They are very promising although their efficient numerical implementations making it competitive with the corresponding parent correlated wavefunction-based methods have not been developed yet. 

Ab initio MP2/6-311++G(d,p) calculations were performed to obtain geometries and electron densities. The perturbative MP2 approach is one of the wavefunction-based methods most frequently used to include electron correlation and its well-gained reputation needs no further comments. The basis set... 

Comparing parameters from a AI results with such from highly resolved spectroscopic data is rather stimulating for the validation and further improvements (developments) of wavefunctions based methods. 

However, care has to be taken when applying DFT to hydrocarbon species in zeolites. The currently available functionals do not properly account for dispersion, which is a major stabilizing contribution for hydrocarbon-zeolite interactions. Due to the size of the systems it is difficult to apply wavefunction-based methods such as CCSDfT) or MP2. Thanks to an effective MP2/DFT hybrid approach and an extrapolation scheme energies, including the dispersion contribution, are now available for the different hydrocarbon species of Fig. 22.1 [50]. 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

In the solution of the electronic Schrodinger equation via wavefunction-based methods, there are two major sources of error fhaf musf be considered. One is fhe expansion of fhe many-elecfron wavefuncfion in terms of Slafer deferminanfs (fhe "mefhod") and fhe ofher is fhe represenfafion of the 1-particle orbitals by a suitable basis set, typically consisting of Gaussian-fype functions, from which fhese defer-minants are constructed. In general, similar considerations also apply in common implementations of density functional theory (DFT), however the first approximation then involves the chosen form of the correlation and exchange functionals. In any event, each of these two expansions, except in very special cases, are necessarily incomplete and they separately impact the final accuracy of an electronic... 

To conclude, we have presented our (fairly subjective) view of what tools are available nowadays to compute electronic couplings for charge transfer processes. We have surveyed in detail the FDE-ET method simply because we are among the developers of this method. Other methods based on DFT, and those that are best suited for being coupled with wavefunction based methods have also been discussed. The discussion also touches on the strengths and limitations of the various methods. 

Within the so-called wavefunction-based methods, Eq. (5) is most often solved by first approximating Wg as a single Slater determinant. Thereby correlation effects are by definition ignored. (Parts of) these may, however, be added subsequently either directly or via perturbation theory. The N single-particle functions 0i, 02, , of the Slater determinant are calculated by solving the Hartree-Fock single-particle equations... 

Potential energy landscape for proton transfer in (H20)3H+ comparison of density functional theory and wavefunction-based methods "... 

The theoretical determination of cluster stability requires some care. The HF method severely underestimates the cluster binding energy. Correlation accounts for about 50 % of the stability in Li clusters and 90 % of the stability in Na and K clusters. [25] This makes it difficult, if not impossible, to treat larger aggregates (especially of transition metals) by means of wavefunction based methods. Alternatives are provided by density functional methods or, with some caution, by well... 

Traditional quantum chemistry starts from the electronic Schrodinger equation (SE) and attempts to solve it using increasingly more accurate approaches (Hartree-Fock and different post-Hartree-Fock methods, see Chapters 4 and 5). These approaches are based on the comphcated many-electron wavefunction (and are therefore called wavefunction-based methods) and in these ab-initio methods no semiem-pirical parameters arise. Such an approach can be summarized by the following se-... 

One of the original approximate methods is the wavefunction-theory-based Hartree-Fock (HF) method [40]. The HF method is a single determinant method that does not include any correlation interactions between the electrons, and as such has limited accuracy [41, 42]. Higher level wavefunction-based methods such as coupled cluster [43 5], configuration interaction [40,46,47], and complete active space [48-50] methods include multiple determinants to incorporate some of the electron-electron correlation. Methods based on perturbation theory, such as second order Mpller-Plesset perturbation theory [51], go beyond the HF method by perturbatively adding electron correlation. These correlated wavefunction-based methods have well-defined ways in which they approach the exact solution to the Schrodinger equation and thus have the potential to be extremely accurate, but this accuracy comes at a price [52]. 

In the following we will briefly revise the key methods that can be employed to study electronic excitations and calculate photoabsorption or photoelectron spectra. However, the focus of this section will be on the description of time-dependent DFT, which can be counted for the most popular method to calculate excitations in large molecules and clusters. Wavefunction-based methods will be discussed oifly very briefly and not exhaustively here. Additionally, we will also make the connection to the experimental measured photoabsorption cross section. As we will state below, this connection is often a bit neglected, but nevertheless important. 

---