Chapter 2
Molecular Modeling and Optimization
Abstract
Recently theoretical quantum chemical calculations have become complementarities for experimental methods in many fields. Therefore our objective is to apply quantum chemical calculations to investigate the fundamental nature of the ionic liquid (IL)–sulphur–nitrogen systems at atomic and molecular level. The first step in this direction is to introduce the readers to a brief overview of the ab initio quantum chemical calculations. It presents the different levels of theories, basis sets, and the working of quantum chemistry programs. The procedure for geometry optimization for ILs is also outlined briefly.
Keywords
Density functional theory; Geometry optimization; Ionic liquids; Quantum chemistry
2.1. Ab Initio Methods
Quantum theory is based on the Schrödinger equation:
? describes the state of the system as a function of coordinates. This function, called the state function or wave function, contains all of the information that can be determined about the system. ˆ is the Hamiltonian (i.e., energy) operator of the system, and E is the energy of that particular state. The Schrödinger equation for molecular systems can only be solved approximately. The approximation methods can be categorized as either ab initio or semi-empirical. Semi-empirical methods use parameters that compensate for neglecting some of the time-consuming mathematical terms in the Schrödinger equation, whereas ab initio methods include all such terms. The term ab initio implies a rigorous, nonparameterized molecular orbital (MO) treatment derived from first principles. However, this is not completely true. There are several simplifying assumptions in ab initio theory, but the calculations are more complete, and therefore more expensive, than those of the semi-empirical methods. It is possible to obtain chemical accuracy via ab initio calculations, and this approach is especially favored in situations in which little or no experimental information is available.
Ab initio theory makes use of the Born–Oppenheimer (
Born and Oppenheimer, 1927) approximation that the nuclei remain fixed on the time scale of electron movement; that is, that the
electronic wave function is unaffected by nuclear motion. It also assumes that
basis sets adequately represent MOs. Each MO (one electron function)
i is expressed as a linear combination of
n basis functions F
m.
The coefficients cmi are called MO expansion coefficients or simply MO coefficients. These basis functions are usually located at the center of atoms and are therefore often called atomic basis functions. The basis functions used in MO calculations are usually described through an abbreviation or acronym such as “6-31G(d).”
2.2. Basis Sets
A basis set is essentially a finite number of
atomic-like functions over which the MO is formed via
linear combination of atomic orbitals (LCAO) methods, as summarized in
Eqn (2.2).
The first stage of all ab initio calculations is a single-determinant SCF (self-consistent field) calculation. Its quality depends on the basis set (i.e., the LCAO used for the calculation and the computational method used).
Equation (2.1) is solved assuming the electron to move in a field of “fixed” electrons and nuclei. First, a set of trial solutions (?) is obtained, which is used to calculate the Coulomb and Exchange operators. The Hartree–Fock equations are then solved giving a second set of solutions ?, which is used in the next iteration. This approach (i.e., SCF) continues until the energies for all of the electrons remain unchanged. The object of all MO programs is to build a set of MOs to be occupied by the electrons assigned to the molecule. In principle, this can be achieved by combining any number of different types of electron probability functions, or even by writing one extremely complex function to describe the electron density in each MO. A far more convenient way is to build up the MOs from sets of orbitals centered on the constituent atoms. The MO calculation then simply involves finding the combinations of these atomic orbitals that have the proper symmetries and that give the lowest (most negative) electronic energy. This is the LCAO formalism.
There are many different possible choices of atomic orbitals (the
basis set). The
Slater-type orbitals (
Slater, 1930) (STOs) have the following form in Cartesian coordinates:
abcSTO(x,y,z)=Nxaybzce-?r
The
Gaussian-type orbitals (
Boys, 1950) (GTOs) have the following form in Cartesian coordinates:
abcGTO(x,y,z)=Nxaybzce-?r2
N is the normalization constant, and a, b, and c are non-negative integers that control the angular momentum L=a+b+c. When a+b+c=0 (i.e., a=0, b=0, c=0), the Gaussian is called an s-type Gaussian; when a+b+c=1, we have a p-type Gaussian, which contains the factor x, y, or z. ? controls the width of the orbital—large ? gives a tight function and small ? gives a diffuse function. Almost all modern ab initio calculations use GTO basis sets. These bases, in which each atomic orbital is made up of several Gaussian probability functions, have considerable advantages over other types of basis sets for the evaluation of one- and two-electron integrals. For instance, they are much faster computationally than the equivalent Slater orbitals. The Gaussian series of programs deals exclusively with GTOs and includes several optional GTO basis sets of varying size. This is one of the main advantages of such a widely distributed program system—the methods and basis sets used become standard and a direct comparison with literature data is often possible.
The simplest of the optional basis sets in
Gaussian03 (
Frisch et al., 2004) is the STO-3G (
Collins et al., 1976;
Hehre et al., 1969). STO-3G is an abbreviation for
Slater-
Type-
Orbitals
simulated by
3 Gaussian functions each. This means that each atomic orbital consists of three Gaussian functions added together. The coefficients of the Gaussian functions are selected so as to give as good of a fit as possible to the corresponding STOs. STO-3G is a
minimal basis set. This means that it has only as many orbitals as necessary to accommodate the electrons of the neutral atom. Because a complete basis set of
p orbitals must be added to maintain spherical symmetry, the elements boron to neon each have five atomic orbitals: 1
s, 2
s, 2
px, 2
py, and 2
pz; for beryllium and lithium, a minimal basis set actually requires only 1
s and 2
s orbitals. However, in STO-3G, the three 2
p orbitals are also included for these elements to give a consistent description across the periodic table. Because there is only one best fit to a given type of Slater orbital (1
s, 2
p, etc.) for each number of Gaussian functions, all STO-3G basis sets for any row of the periodic table are identical except for the
exponents ? of the Gaussian functions. These are expressed as a
scale factor, the square of which is used to multiply all exponents in the original best-fit Gaussian functions. In this way, the ratios of the exponents of the individual Gaussians to each other remain constant, but the effective exponent of the entire orbital can be varied. The exponents, or scale factors, can be considered to be a measure of the extent of the orbital. A low exponent indicates a diffuse (and therefore relatively high-energy) orbital; high exponents indicate compact orbitals close to the nucleus.
The STO-3G basis set is very economical, having only one basis function (or atomic orbital) per hydrogen atom (the 1s), five per atom from Li to Ne (1s, 2s, 2px, 2py, and 2pz), and nine per atom for the second-row elements Na to Ar (1s, 2s, 2px, 2py, 2pz,...
