Wave Function Based Methods

The key idea behind solving the electronic Schrodinger eqnation is to start with an initial guess for the total electronic wave fnnction with some freely adjustable parameters, and vary these parameters nntil the lowest energy is found. The variational principle then states that this energy is the ground-state energy. 

The focus in this chapter is on quantum chemical methods. These can be classified as semiempirical, ab initio, and density functional theory (DFT) methods. The latter ones usually involve empirical parameterization and, hence, sometimes are also considered as semiempirical methods. Alternatively, a distinction on the basic quantity - wave function (WF) or electronic density - can be made as wave-function-based methods (WFT) and DFT. In this classification scheme, wave-function-based methods include semiempirical as well as ah initio procedures. Although the impact of semiempirical methods on the progress of quantum chemistry can hardly be overestimated [13], their use now is mainly restricted to very large systems [14]. Thus, in the following the description of wave-function-based procedures will be restricted to ah initio methods. 

The price-performance ratio of DFT is very good and, thus, DFT has dominated applied theoretical chemistry during the last few decades. However, some serious shortcomings, even in seemingly well-behaved systems, for example, simple alkane isomerization reactions [15], have led to a renewed interest in wave-function-based methods [16]. 

The basic quantity of WF-based electronic structure methods, namely the wave function, can be obtained as solution of the time-independent, nonrela-tivistic SchrSdinger equation. In addition, the vahdity of the Born-Oppenheimer approximation (separabiUty of nuclear and electronic motion) is assumed. The electronic SchrSdinger equation is then solved for fixed nuclei in other words, the nuclear coordinates are parameters rather than variables in the wave function. The electronic Hamiltonian contains pairwise electron-electron interaction energies, meaning that the motion of the individual electrons is not independent of each other but is correlated.  

A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated. Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to compute the electron density, not the electron wave function. There are many methods, however, where the object of the calculation is to compute the full electron wave function. These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that, given infinite computer time, can converge to the exact solution of the Schrodinger equation. We cannot do justice to the breadth of this field in just a few paragraphs, but several excellent introductory texts are available 

Before giving a brief discussion of wave-function-based methods, we must first describe the common ways in which the wave function is described. We mentioned earlier that the wave function of an /V-particle system is an tV-dimension al function. But what, exactly, is a wave function Because we want our wave functions to provide a quantum mechanical description of a system of N electrons, these wave functions must satisfy several mathematical properties exhibited by real electrons. For example, the Pauli exclusion principle prohibits two electrons with the same spin from existing at the same physical location simultaneously. We would, of course, like these properties to also exist in any approximate form of the wave function that we construct. 

Given the complexity of the systems and the diversity of the questions still open in the field of metal clusters, it is no wonder that essentially all the methods available from the ample arsenal of quantum chemistry have been applied to cluster problems. We will not give an extensive overview of the many different methods (let alone aim for completeness) and leave aside most technical aspects. This information can be found in spedalized publications (e. g. [11-15]), from which some are even devoted to the electronic structures of clusters. [16, 17] Instead, we will summarize the basic features of the methods and comment on their applicability to the description of both naked and ligated metal dusters. We will start the discussion with wave function based methods and then proceed to density functional methods. Although the latter have only recently gained a broader acceptance for chemical applications, they have a rich tradition in the metal cluster field, particularly due to their solid state heritage. We will also briefly mention simplified approaches to the electronic structure of metal clusters. 

First prindple quantum chemical methods, whether wave function based ( ab initio ) or density based, are aimed at solving the electronic Schrddinger equation without any reference to adjustable parameters or empirical data. In their standard form, they invoke the Bom-Oppenhdmer separation of electronic and nuclear motion and employ a nonrelativistic Hamiltonian which does not include any explicit reference to spin-dependent terms. Many quantum chemical methods are based on the variational prindple which, for computational convenience, is implemented in algebraic form via either one-electron functions built from linear combinations of atomic orbitals or n-electron functions constructed from Slater determinants. [11, 12] 

The basis for all wave function based ab initio methods is the Hartree-Fock (HF) approach. [11, 12] It makes use of a single-determinant ansatz constructed from one-electron spin orbitals. These orbitals describe the motion of each electron within the field of the nuclei and the mean field of the remaining n-1 electrons. The mean field is not known a priori, but depends on the orbitals which are determined self-consistently from the eigenvalue problem of the Fock operator. 

2 Electronic Structures of Metal Clusters and Cluster Compounds 

In order to improve the mean field description of the electronic structure one has to go beyond the single-configuration approach. [12, 13] IWo main strategies have been developed to introduce correlation effects. In the first case, one employs methods based on many-body perturbation theory (MBPT). [12, 21] They allow the treatment of so-called dynamical correlation effects in cases where the HF method already provides a reasonable description of the ground state. However, these perturbation theoretical methods are not variational, that is the calculated value for the energy does not provide an upper bound to the true energy of the system. 

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It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Before we start looking at possible approximations to Exc we need to address whether there will be some kind of guidance along the way. If we consider conventional, wave function based methods for solving the electronic Schrodinger equation, the quality of the... 

In regular wave function based methods J is determined through the four-center-two-... 

During the last years, more and more researchers have applied density functional theory to small transition-metal complexes and benchmarked the results against either high level wave function based methods or experimental data. A particular set of systems for which reasonably accurate benchmark data are available are the cationic M+-X complexes, where X is H, CH3 or CH2. Let us start our discussion with the cationic hydrides of the 3d transition-metals. 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

In order to use wave-function-based methods to converge to the true solution of the Schrodinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both FIF and post-HF calculations. Even with these improvements, however you can appreciate the problem with... 

This situation changed drastically when it was discovered in the 1990s that density functional (DF) methods do a much better job of modeling force fields than (affordable) wave function based methods. Already within the local density approximation (LDA) of DF theory, vibrational frequencies were predicted with... 

In contrast to molecular mechanics force fields, modern semiempirical methods are classified as an SCF electron-structure theory (wave function-based) method [8]. Older (pre-HF)... 

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