Читать книгу Computational Methods in Organometallic Catalysis - Yu Lan - Страница 20

The core requirement of quantum chemistry is the solution of the time‐independent Schrödinger equation

where Ĥ is the Hamiltonian operator, Ψ is the wavefunction for all of the nuclei and electrons, and E is the energy associated with this wavefunction. The Hamiltonian contains all operators that describe the kinetic and potential energy of the molecule at hand. Schrödinger equation is the basis of quantum mechanics, which was proposed by E. Schrödinger, an Austrian theoretical physicist, in 1926 [1–3]. It describes the law of the state of microparticles changing with time. The state of microsystem can be described by wavefunctions, whose differential equation is Schrödinger equation. It means that the wavefunctions can be solved by the equation, when the initial conditions and boundary conditions are given.

The Hamiltonian operator Ĥ can be broken into two operators

Those two operators represent kinetic energy (T) and potential energy (V), respectively. It could be further broken according to the nuclei and electron parts as

which represent (i) kinetic energy of nuclei, (ii) nuclear–nuclear repulsions, (iii) kinetic energy of electrons, (iv) nuclear–electron attraction, and (v) electron–electron repulsion. In most cases, it is impossible and unnecessary to find an analytic solution for the existing Schrödinger equation.

For organic system, several assumptions are indispensable to arrive at a solution due to the excessive degree of freedom. One of the most important assumptions is the Born Oppenheimer approximation proposed by M. Born and R. Oppenheimer in 1927 [4]. Considering that the nucleus is much larger than the electron generally by 3–4 orders, the moving of the electron should be much more rapid than that of the nucleus under the same interaction. The result of this difference in velocity is that the electron moves at every moment as if it were in a potential field composed of stationary nucleus, while the nucleus cannot observe the specific position of electrons and can only be averaged interactions. Thus, the variables of nuclear coordinates and electronic coordinates can be approximately separated. The complex process of solving the whole‐wave function can be decomposed into two relatively simple processes, solving the electronic wavefunction and solving the nuclear wavefunction

This separation of the total wavefunction into an electronic wavefunction ϕ(r) and a nuclear wavefunction Ф(R) means that the positions of the nuclei can be fixed and then one only has to solve the Schrödinger equation for the electronic part. Usually, we only focus on the electron energy of the potential energy surface, which is determined by the electronic wavefunction ϕ(r). Therefore, the Hamiltonian obtained after applying the Born–Oppenheimer approximation and neglecting relativity is

where V^nuc is the nuclear–nuclear repulsion energy.

In 1930, Hartree–Fock (HF) theory was formulated by V. Fock and D. R. Hartree, which is the basis of any other methods for solving Schrödinger equation [5, 6]. The core of HF theory is to simplify the problem of solving the multiparticle system in the external field into a problem of solving the wavefunction of a single particle. The electronic wavefunction can be separated into a product of functions that depend only on one electron

Unfortunately, the effect of electron–electron repulsion cannot be solved; therefore, the Schrödinger equation still cannot be solved exactly. Alternatively, the exact electron–electron repulsion is replaced with an effective field V_i^eff produced by the average positions of the remaining electrons. With this assumption, the separable functions ϕ_i satisfy the Hartree equations

In the above‐mentioned equation, solving for a set of functions ϕ_i is still problematic because the effective field V_i^eff is dependent on the wavefunctions. To solve this problem, an iterative procedure, named self‐consisted field (SCF), was proposed by D. R. Hartree in 1927. First, a set of functions (ϕ₁, ϕ₂, …, ϕ_n) is assumed, which can be used to produce the set of effective potential operators V_i^eff and the Hartree equations are solved to produce a set of improved functions ϕ_i. These new functions produce an updated effective potential, which, in turn, yields a new set of functions ϕ_i. This procedure is repeated until the functions ϕ_i no longer change (converge) and produce an SCF. SCF convergence is a necessity for energy calculations.

To further simplify approximation of wavefunctions, linear combination of atomic orbitals (LCAO) theory was proposed by Roothaan in 1951 [7, 8]. A LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In a mathematical sense, these wavefunctions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.

An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense, n atomic orbitals combine to form n molecular orbitals, which can be numbered i = 1 to n and which may not all be the same. The linear expansion for the ith molecular orbital would be

where φ_i is a molecular orbital represented as the sum of n atomic orbitals χ_r, each multiplied by a corresponding coefficient c_ri, and r (numbered 1 to n) represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The Hartree–Fock procedure is used to obtain the coefficients of the expansion. The orbitals are thus expressed as linear combinations of basis functions, and the basis functions are one‐electron functions, which may or may not be centered on the nuclei of the component atoms of the molecule. In either case, the basis functions are usually also referred to as atomic orbitals, which are typically those of hydrogen‐like atoms since these are known analytically, i.e. Slater‐type orbitals (STOs), but other choices are possible such as the Gaussian‐type functions from standard basis sets or the pseudo‐atomic orbitals from plane‐wave pseudopotentials.