Spherical-harmonic GTOs

An alternative solution to the gauge-independence problem in molecular calculations is to attach the complex phase factors directly to the atomic basis functions or atomic orbitals (AOs) rather than to the MOs. Thus, each basis function—which in modern calculations usually corresponds to a Gaussian-type orbital (GTO)—is equipped with a complex phase factor according to Eq. 87. A spherical-harmonic GTO may then be written in the from... 

Completeness of the single-exponent spherical-harmonic GTOs follows from the fact that the HO functions (which constitute a complete set) may be written as finite linear combinations of GTOs, for example... 

Equipped with a complete set of simple spherical-harmonic GTOs of the form (6.6.6), we now turn our attention to the description of the radial space by means of variable exponents. We begin by comparing the expansion of the radial part of the one-electron space by means of variable exponents and by means of the principal quantum number n. 

Complex algebra is avoided by rewriting the spherical harmonics in terms of the real solid harmonics of Section 6.4.2 (which then absorb the monomials in r), leaving us with the following simple set of real-valued spherical-harmonic GTOs... 

We have finally arrived at a set of basis functions suitable for nmlecular calculations the spherical-harmonic GTOs in the form (6.6.15) with variable exponents for the radial part of the... 

Before we begin our discussion of Gaussian basis sets, let us briefly review the one-electron basis functions studied in Chapter 6. The complex spherical-harmonic GTOs are given by... 

As discussed in Chapter 8, the primitive Cartesian GTOs (9.1.3) are mostly used in fixed linear combinations X/i(r). A typical AO thus consists of a linear combination of primitive Cartesian GTOs of the same angular-momentum quantum number / but of different Cartesian quantum numbers i, j and k and of different exponents a. In Section 9.1.2, we shall discuss how Cartesian GTOs of the same / but different i, j and k are combined to yield the real-valued herical-harmonic GTOs next, in Section 9.1.3, we shall see how the GTOs of different exponents are combined to yield the final AOs as contracted spherical-harmonic GTOs. 

A real-valued spherical-harmonic GTO of quantum numbers / and m. with exponent a and centred on A is given by the expression... 

The primitive Cartesian Gaussians are combined not only in their angular parts, but also in their radial parts. The final contracted GTOs may be written as linear combinations of primitive spherical-harmonic GTOs of different exponents... 

As should be apparent from our discussion so far, a large number of integrals over primitive Cartesian GTOs (9.1.3) contribute to a smaller number of integrals over contracted spherical-harmonic GTOs (9.1.13). This is especially true for the two-electron integrals (9.1.2), since the... 

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