Scheme 1.2. Molecular Polarizability a
1.2. Molecular Orbital Theory and Methods
1.2.3. Ab Initio Methods
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CHAPTER 1 Chemical Bonding and Molecular Structure
Hückel’s rule also pertains to charged cyclic conjugated systems. The cyclo- propenyl (2electrons), cyclopentadienyl anion (6electrons), and cycloheptatrienyl (tropylium) cation (6electrons) are examples of stabilized systems. We say much more about the relationship between MO configuration and aromaticity in Chapter 9.
–
+
+
33
SECTION 1.2 Molecular Orbital Theory and Methods
Specific ab initio methods are characterized by the form of the wave function and the nature of the basis set functions that are used. The most common form of the wave function is the single determinant of molecular orbitals expressed as a linear combination of basis functions, as is the case with semiempirical calculations. We describe alternatives later in this section. Early calculations were often done with Slater functions, designated STO for Slater-type orbitals. Currently most computations are done with Gaussian basis functions, designated by GTOs. A fairly accurate represen- tation of a single STO requires three or more GTOs. This is illustrated in Figure 1.12, which compares the forms for one, two, and three GTOs. At the present time most basis sets use a six-Gaussian representation, usually designated 6G. The weighting coefficients for theN components of a STO-NG representation are not changed in the course of a SCF calculation.
A basis set is a collection of basis functions. For carbon, nitrogen, and oxygen compounds, a minimum basis set is composed of a 1s function for each hydrogen and 1s, 2s, and three 2pfunctions for each of the second-row atoms. More extensive and flexible sets of basis functions are in wide use. These basis sets may have two or more components in the outer shell, which are calledsplit-valencesets. Basis sets may includepfunctions on hydrogen and/ordandf functions on the other atoms. These are called polarization functions. The basis sets may also include diffuse functions, which extend farther from the nuclear center. Split-valence bases allow description of tighter or looser electron distributions on atoms in differing environments. Polarization permits changes in orbital shapes and shifts in the center of charge. Diffuse functions allow improved description of the outer reaches of the electron distribution.
Pople developed a system of abbreviations that indicates the composition of the basis sets used in ab initio calculations. The series of digits that follows the designation 3G or 6G indicates the number of Gaussian functions used for each successive shell.
The combination of Gaussian functions serves to improve the relationship between electron distribution and distance from the nucleus. Polarization functions incorporate additional orbitals, such aspfor hydrogen anddand/orf for second-row atoms. This permits changes in orbital shapes and separation of the centers of charge. The inclusion of dand f orbitals is indicated by the asterisk ∗. One asterisk signifies dorbitals on second-row elements; two asterisks means that p orbitals on hydrogen are also included. If diffuse orbitals are used they are designated by a plus sign+, and the
Fig. 1.12. Comparison of electron distribution for STO, G, 2G, and 3G expressions of orbitals.
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CHAPTER 1 Chemical Bonding and Molecular Structure
designation double plus++means that diffuse orbitals are present on both hydrogen and the second-row elements. Split-valence sets are indicated by a sequence defining the number of Gaussians in each component. Split-valence orbitals are designated by primes, so that a system of three Gaussian orbitals would be designated by single, double, and triple primes (,, and). For example 6-311+ G(d, p) conveys the following information:
•6: Core basis functions are represented as a single STO-6G expression.
•311: The valence set is described by three sets of STO-NG functions; each set includes an s orbital and three porbitals. In the 6-311+G(d) basis there are three such sets. One is composed of three Gaussians (STO-3G expression of one s-type and threep-type forms) and the other two are represented by a single Gaussian (STO-1G) representation of thes-p manifold. The collection of components of the split-valence representation can be designated by a series of primes.
• +: A STO-1G diffuse s-p manifold is included in the basis set for each nonhydrogen atom;++implies that diffuse functions are also included for the hydrogen atoms.
•p: A set of pfunctions placed on each nonhydrogen atom and specifies the composition.
•d: A set of STO-1Gd-functions is placed on each nonhydrogen atom for which dfunctions are not used in the ground state configuration. Ifdfunctions are so used, polarization is effected by a manifold off functions.
The composition of several basis sets is given in Table 1.9.
An important distinguishing feature among ab initio calculations is the extent to which they deal withelectron correlation. Correlation is defined as the difference between the exact energy of a molecular system and the best energy obtainable by a SCF calculation in which the wave function is represented by a single determinant. In single-determinant calculations, we consider that each electron experiences an averaged electrostatic repulsion defined by the total charge distribution, a mean fieldapproxi- mation. These are calledHartree-Fock(HF) calculations. Correlation corrections arise from fluctuations of the charge distribution. Correlations energies can be estimated by including effects of admixtures of excited states into the Hartree-Fock determinant.
Table 1.9. Abbreviations Describing Gaussian Basis Setsa
Designation H C Functions on second-row atoms
3-21G 2 9 1s; 2s, 3 2p; 2s, 3 2p
3-21+G 2 13 1s; 2s, 3 2p; 2s, 3 2p; 2s+, 3 2p+ 6-31G∗or 6-31G(d) 2 15 1s; 2s, 3 2p; 2s, 3 2p; 5 3d
6-31G∗∗or 6-31(d,p) 5 18 1s; 2s, 3 2p; 2s, 3 2p; 2s; 3 2p; 5 3d 6-31+G∗or 6-31+d 3 19 1s; 2s, 3 2p; 2s, 3 2p; 5 3d; 2s+3 2p+ 6-311G∗∗or 6-311(d,p) 6 18 1s; 2s, 3 2p; 2s, 3 2p; 2s; 3 2p; 5 3d 6-311G(df,p) 6 25 1s; 2s, 3 2p; 2s, 3 2p; 2s; 3 2p; 5 3d; 7 4f 6-311G(3df,3pd) 17 35 1s; 2s, 3 2p; 2s, 3 2p; 2s3p; 5 3d, 7 4f a. From E. Lewars,Computational Chemistry, Kluwer Academic Publishers, Boston, 2003, pp. 225–229.
35
SECTION 1.2 Molecular Orbital Theory and Methods
This may be accomplished by perturbation methods such as Moeller-Plesset(MP)51 or by including excited state determinants in the wave equation as inconfigurational interaction(CISD)52calculations. The excited states have electrons in different orbitals and reduced electron-electron repulsions.
The output of ab initio calculations is analogous to that from HMO and semiem- pirical methods. The atomic coordinates at the minimum energy are computed. The individual MOs are assigned energies and atomic orbital contributions. The total molecular energy is calculated by summation over the occupied orbitals. Several schemes for apportioning charge among atoms are also available in these programs.
These methods are discussed in Section 1.4. In Section 1.2.6, we illustrate some of the applications of ab initio calculations. In the material in the remainder of the book, we frequently include the results of computational studies, generally indicating the type of calculation that is used. The convention is to list the treatment of correlation, e.g., HF, MP2, CISDT, followed by the basis set used. Many studies do calculations at several levels. For example, geometry can be minimized with one basis set and then energy computed with a more demanding correlation calculation or basis set.
This is indicated by giving the basis set used for the energy calculation followed by parallel lines (//) and the basis set used for the geometry calculation. In general, we give the designation of the computation used for the energy calculation. The infor- mation in Scheme 1.3 provides basic information about the nature of the calculation and describes the characteristics of some of the most frequently used methods.