3 Theoretical background
3.1 Schrödinger equation
If the electron distribution in materials or molecules is of central importance, as it is in chemical reactions, there is no substitute for quantum mechanics. Electrons are very light particles, that cannot be described even qualitatively correct by classical mechanics. The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics. It describes the temporal evolution of a state of a physical system.
The time-independent non-relativistic Schrödinger equation of a many-body system can be written as:
(
R,r) (
Ψ R,r)
=EΨ(
R,r)
H , (3.1)
where E is the total energy of the system and Ψ is the wave function of the 3n electronic coordinates, r, and the 3N coordinates of the nuclei, R. N is the number of nuclei and n is the number of electrons in the system. H is the many-body Hamiltonian operator. The only fundamental interactions of concern, in solid state physics and most of quantum chemistry, are the electrostatic interactions. In principle, relativistic effects and magnetic effects should be included, but for simplicity these effects are not considered in this work. The Hamiltonian operator for a system of n electrons and N nuclei can be written as:
el respectively, and are given by:
∑
nuclei coordinates, at position RI and the i-th electron coordinates, at position ri, while MI and mi denote the nuclei and electronic masses. ħ is the Planck’s constant divided by 2π. The repulsive Coulomb interactions between the nuclei are represented by∑ ∑
−where ZI and ZJ are the atomic number of nuclei I and J, e is the charge on the electron and ε0 is the vacuum permittivity. The electrostatic potential energy due to the interactions between electrons and nuclei and the repulsion between electrons are given by: eigenfunctions, Ψ, for a given molecule, each characterized by a different (or equivalent in the case of degenerate eigenfunctions) associated eigenvalue E. The whole physical information except for the symmetry of the wave functions is contained in the Hamiltonian operator. The only further information needed are the appropriate quantum statistics and, especially for heavier elements relativistic corrections. Heavy elements have very localized wave functions for the core electrons. As a result, the core electrons have a very high kinetic energy and move with a velocity close to the speed of light. Since chemical reactions involve usually just a change in the valence electron distribution, relativistic effects are often, as a first approximation, neglected for reactions involving atoms up to krypton (36). The heaviest element studied in this work is iron (26). While relativistic effects are present in systems containing iron these Born-Oppenheimer approximation, the introduction of one-electron equations, the concept of orbitals, and the basis sets used to construct them are common to most ab initio methods and will be described in the next sections.
An introduction to quantum mechanics and solutions of the Schrödinger equation can be, for example, found in the books of Levine (2000), Sakurai (1994) and
Cohen-Tannoudji, Diu and Laloë (1977). Further details of the implementation of electronic structure theory are given e.g. in the books of Szabo and Ostlund (1996), Yarkony (1995) and Helgaker et al. (2002). Elementary review articles for chemical engineers are written by Keil (2004) and Santiso and Gubbins (2004).
3.1.1 Born-Oppenheimer approximation
One difficulty in solving Eqn. (3.1) arises from the large number of variables the many-body wave function, Ψ, depends on. For a system consisting of n electrons and N nuclei there are 3n+3N degrees of freedom, i.e. three spatial coordinates for each electron and for each nucleus. A first simplification can be achieved by taking into account the large difference in masses between the electrons and the nuclei (nuclei are approximately 103 to 105 times heavier than electrons). Since the nuclei are much heavier than the electrons, their motion will be much slower, i.e. the characteristic time scales of processes involving the electrons are much smaller. Hence, it is supposed that the electrons follow the motion of the nuclei instantaneously. In other words, the electronic wave functions can be found by assuming that the nuclei are fixed in space.
The nuclei coordinates are just parameters in the electronic Schrödinger equation and as a first approximation the motions of the electrons and nuclei are decoupled. As a result, the full Hamiltonian is split. The electronic Hamiltonian, Hel, for fixed nuclear coordinates, R, can be written as
el el el nucl el
el T V V
H = + − + − (3.6)
and the Schrödinger equation for the electrons, for a fixed given configuration of the nuclei, is
el el el
el E
H ψ = ψ (3.7)
The nuclei, in turn, are assumed to move according to the atomic Schrödinger equation:
(
Tnucl +Vnucl−nucl +Eel)
ψnucl = Enuclψnucl (3.8) The potential energy surface (PES) or Born-Oppenheimer (BO) energy surface,el nucl nucl
BO V E
V = − + , (3.9)
is taken to be the potential energy for the nuclear motion. Strictly speaking, this nuclear motion should be treated quantum mechanically. In practice, it is sufficient to solve a classical equation of motion for the nuclei, as quantum-mechanical effects, such as zero point vibrations or tunneling, can be corrected for in an ad hoc way if necessary. To summarize, within the BO approximation first the electronic eigenvalue equation, Eqn.
(3.7), has to be solved, and then Eqn. (3.9) is applied to obtain the potential energy surface (PES). It is noted that the eigenvalue equation, Eqn. (3.7), has an infinite number of solutions. In many cases one is just interested in the solution with the lowest energy, which corresponds to the ground state of the electronic system.
Most ab initio methods make use of the here described Born-Oppenheimer (and adiabatic) approximation, proposed by Born and Oppenheimer in 1927. A detailed mathematical description of the Born-Oppenheimer and adiabatic approximation can e.g. be found in the books from Jensen (1999) and Hirst (1985). It is important to note that the adiabatic and Born-Oppenheimer approximation breaks down when two or more solutions of the electronic Schrödinger equation come energetically close together.
If non-adiabatic effects are important it is possible to include corrections after a BO calculation has been made using surface-hopping methods developed by Tully (Preston (1971), Tully (1971) and Tully (1990)). Alternatively, very sophisticated quantum scattering or semiclassical approaches have to be used. Section 3.3 discusses non-adiabatic effects and how they are dealt with approximately in this work.
3.1.2 Variational principle and molecular orbitals
Many ab initio methods are based on the Rayleigh-Ritz variational principle. Given any normalized electronic wave function, Φe(r), which need not be the true solution to the Schrödinger equation, Eqn. (3.7), the expectation value of the Hamiltonian:
[ ]
Φe =∫
d Φ HΦeE r *e , (3.10)
corresponding to Φe(r) will always be greater than or equal to the ground state energy E0 of the electronic Schrödinger equation, i.e.:
[ ]
E0EΦe ≥ for all Φe. (3.11)
Since the Hamiltonian is a Hermitian operator the proof of this principle is simple and can be found in any elementary quantum chemistry text book. Hence, finding the ground state energy and wave function of a many-body system can be formulated as a variational minimization problem:
[ ]
Φ0 =0δE . (3.12)
The wave function appearing in the electronic problem (3.7) depends on the coordinates of all electrons. As a next approximation the trial molecular wave functions are chosen as combinations of single electron functions, so called Slater determinants, see Slater (1929): function is antisymmetric upon electron interchange and obey the Pauli exclusion principle that in a system of identical fermions (e.g. a multi-electron system) two particles can never occupy the same quantum state. The spatial orbitals are usually expressed in terms of a set of basis functions φµ(r):
( ) ∑ ( )
These basis functions are often (and always in this work) centered at the nuclei. When functions centered at different nuclei in a molecule are combined linearly as in Eqn.
(3.14), the spatial orbitals are called LCAO-MO (Linear Combinations of Atomic Orbitals - Molecular Orbitals). Once a set of basis functions is chosen, the “best”
electronic wave function within the basis is obtained by minimizing the energy with respect to the expansion coefficients Cµi. The shape and kind of the basis functions used in this work will be described in section 3.2.4.
3.1.3 Hartree-Fock method
The idea to replace the complicated many-electron problem (Eqn. 3.7) by a one-electron problem in which electron-electron repulsion is treated in an average way is called Hartree-Fock (HF) approximation. The Hartree-Fock method (see Hartree (1928) and Fock (1930)) is based on the following assumptions: 1) the Born-Oppenheimer approximation, 2) the many-electron Hamiltonian is replaced by a sum of effective one-electron Hamiltonians which act on one-one-electron wave functions (a single Slater determinant) and 3) the Coulomb repulsion between electrons is represented in an averaged way. The matrix elements of the effective Hamiltonian depend on the wave function, i.e. the Schrödinger equation becomes non-linear and requires a self-consistency procedure. All self-consistent mean-field (SCF) theories like the Hartree-Fock method lead to equations of the form:
( ) (
i χ i,σ)
εχ(
i,σ)
f r = r , (3.15)
where the Fock operator f(i) can be written as:
( ) ( )
ii m
f i eff
i
υ +
∇
−
= 2
2
2
h . (3.16)
Here ri are the spatial coordinates of the ith electron, χ are the spin orbitals and υeff is the effective potential “seen” by the ith electron which depends on the spin orbitals of the other electrons. Using an atomic orbital (AO) basis the so-called Roothaan matrix equations (Roothaan (1951)) have to be solved iteratively in the SCF procedure for closed shell molecules, where all electron spins are paired:
k k
k SC
FC =ε . (3.17)
For unrestricted open shell calculations separate spatial orbitals for α and β spin are used and the Pople-Nesbet equations (Pople and Nesbet (1954)) have to be solved:
α α α
αCk εkSCk
F = and FβCkβ =εkβSCβk (3.18)
Here F is the Fock matrix, C is a square matrix of molecular orbital coefficients, S is the overlap matrix and εεεε is the diagonal matrix of the orbital energies. The larger and more complete the set of basis functions in the Hartree-Fock method is, the greater is the degree of flexibility in the expansion for the spin orbitals and the lower will be the expectation value of the ground state energy. The lowest energy value achievable in this way is called the Hartree-Fock limit, EtotHF.
The representation of the wave function by a single Slater determinant function includes Fermi statistics (“exchange” effects), but does not account for all correlation effects.
Electronic screening and instantaneous electron-electron correlations are not described properly. Electron correlation effects, Ec =Etot −EtotHF, are commonly included in post HF methods, such as Møller-Plesset perturbation theory, configuration-interaction (CI) or coupled-cluster (CC) theory. Such methods are, however, computationally very demanding and are currently limited to a rather small number of atoms and are not applied in this work. A good review of ab initio methods for electron correlation in molecules can e.g. be found by Knowles et al. (2000).