One-particle basis functions

The majority of quantum-chemistry calculations have been carried out by employing the independent particle model in the framework of the HF method. In the most widely used approach molecular orbitals are expanded in predefined one-particle basis functions which results in recasting the integro-differential HF equations into their algebraic equivalents. In practice, however, the basis set used is never complete and very often far too limited to describe essential features of HF orbitals, for example, their behaviour in the vicinity of nuclei. That is why such calculations always suffer from the so called basis set truncation error . This error is difficult to estimate and often leads to low credibility of the results. 

The one-particle basis functions x are often referred to as atomic orbitals (AOs). The MO coefficients C are obtained by solving an electronic structure problem simpler than that of Eq. (2), such as the independent particle (Hartree-Fock) approximation, or using a multiconfigurational Hartree-Fock approach. This has the advantage that these approximations generally... 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

A practical advantage of the finite-nucleus model is that extremely high exponents of the one-particle basis functions are avoided. Since for quantities of chemical interest it is not very important which nuclear model is actually used, the Gaussian charge distribution is often applied, being the most convenient choice. 

In this chapter, the case of general molecules — meaning molecules of arbitrary structure and thus arbitrary external nuclear potential — is considered. We will understand how the numerical solution of mean-field equations for many-electron atoms helps us to solve the molecular problem. The key element is the introduction of analytically known one-particle basis functions rather than an elaborated numerical treatment on a three-dimensional spatial grid, which is feasible but not desirable (a fact that will become most evident in section 10.5). 

Figure 5 Illustration of the spaces spanned by the one-particle basis functions i, ],.. = occupied spin-orbitals, a,b,.. . = non-occupied spin-orbitals contained in the finite basis, a, p,... = complete set of virtuals, p,q,... = arbitrary spin-orbitals contained in the finite basis, 1C, A,... = complete basis set...

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

Many of these points are well illustrated by Cu2, which has become a benchmark for theoretical calculations owing to its relative simplicity and the availability of accurate experimental data. The theoretical spectroscopic constants are quite poor unless the 3d electrons are correlated, even though both Cu atoms nominally have a 3d °4s occupation. In fact, quantitative agreement with experiment is achieved only if both the 3d and 4s electrons are correlated, both higher excitations and relativistic effects are included, and large one-particle basis sets, including several sets of polarization functions, are used (24,25). This level of treatment is difficult to apply even to Cua, let alone larger Cu clusters. 

For W2 theory, we opted for CCSD(T)/VQZ+1 as the level of theory for reference geometries. For geometries, the VQZ basis set is known to be close to the one-particle basis set limit [17, 18], while the addition of the inner polarization functions again takes care of inner polarization effects. 

Thus the x matrix B is a function of B and therefore of r real numbers, which in our approach play the role of the parameters for A-representable 2-matrices within the limitations of the given one-particle basis set. Compare this with the = ( ) parameters of the FCl approach. Recall Kummer s basic theorem [1, Theorem 2.8, p. 56] that B could be a second-order RDM if and only if B ) is a positive operator on A-space. For 2, /i real and 2 > 0, we set... 

The components of vectors D k.=i..m are completely defined by the parameters of the underlying full-CI type wave function, and the index sets of Slater-determinants and their subdeterminants according to (13). The number of vectors D is ( ), and this is of course equal to the number of geminals g constructed over the M-dimensional one-particle basis Bm-... 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

Thus the one-particle basis determines the MOs, which in turn determine the JV-particle basis. If the one-paxticle basis were complete, it would at least in principle be possible to form a complete jV-particle basis, and hence to obtain an exact wave function variationally. This wave function is sometimes referred to as the complete Cl wave function. However, a complete one-paxticle basis would be of infinite dimension, so the one-paxticle basis must be truncated in practical applications. In that case, the iV-particle basis will necessarily be incomplete, but if all possible iV-paxticle basis functions axe included we have a full Cl wave function. Unfortunately, the factorial dependence of the iV-paxticle basis size on the one-particle basis size makes most full Cl calculations impracticably large. We must therefore commonly use truncated jV-paxticle spaces that axe constructed from truncated one-paxticle spaces. These two truncations, JV-particle and one-particle, are the most important sources of uncertainty in quantum chemical calculations, and it is with these approximations that we shall be mostly concerned in this course. We conclude this section by pointing out that while the analysis so fax has involved a configuration-interaction approach to solving Eq. 1.2, the same iV-particle and one-particle space truncation problems arise in non-vaxiational methods, as will be discussed in detail in subsequent chapters. 

Starting from a set of localised pair functions or geminals h = (ht,h 2,...h m) obtained from appropriate pairing of one particle basis spin functions, we will devise the following transformations, the motive to be explained below,... 

Further, there are asymptotically corrected XC kernels available, and other variants (for instance kernels based on current-density functionals, or for range-separated hybrid functionals) with varying degrees of improvements over adiabatic LDA, GGA, or commonly used hybrid DFT XC kernels [45]. The approximations in the XC response kernel, in the XC potential used to determine the unperturbed MOs, and the size of the one-particle basis set, are the main factors that determine the quality of the solutions obtained from (13), and thus the accuracy of the calculated molecular response properties. Beyond these factors, the quality of the... 

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