Restricted open shell Hartree-Fock (ROHF)

The closed shell Hartree-Fock method described in the previous section assumes that spin-orbitals are occupied by two electrons with opposite spin. In an open shell approach, orbitals occupied by single electrons of arbitrary spin are respected in addition to doubly occupied orbitals, resulting in a larger basis function set. This can easily be done using a (larger) Slater determinant of single electron spin orbitals. Thus within the Slater determinant representing the system LCAO in the unrestricted open shell HF approach, individual singly occupied orbitals may appear twice with opposite spin components.

In all other respects the open shell approach is very similar to the closed shell approach [26]. It is important, however, to choose an appropriate number of higher orbitals in the open shell approach, depending on the situation at hand. The drawback of the open shell method is that the Slater determinant in this setting is not necessarily an eigenfunction of the total spin anymore.

(It is in the closed shell approach, where all electronic spins are paired with an electron of opposite spin in all contributing spatial orbitals.) Since the total spin S commutes with the Hamiltonian in equation 3.2, it is important for systems with non-zero total spin to choose Slater determinants which are eigenfunctions of the total spin for calculations using the open shell methods.

An open shell ansatz for the wavefunction is required, which ensures that the corresponding Slater determinants are already eigenfunctions of the total spin.

This ansatz for the wavefunction is called the Restricted Open shell Hartree-Fock (ROHF) method. This is the method used in our O₂-O₂ supermolecule PES calculations, which require an open shell approach due to their non-zero total spin.

The ROHF ansatz starts out from a spinS=0 closed shell wavefunction (all electrons paired with one of opposite spin). Then it is assumed that one single electron of a high orbital is excited from statem to staten, undergoing a spin flip at the same time, yielding a triplet state with a z component of -1.

3−1Ψⁿ_m =|Ψ₁Ψ¯₁· · ·Ψ_gΨ¯_gΨ¯_mΨ¯_n| (3.11) Here, the ¯Ψ spinorbitals have spin down. The n and m indices in ³₋₁Ψ^m_n sym-bolise the excitation from orbitalmton. Eigenstates of the total spinS=1 can now be obtained by application of the spin operator S₊, which raises the spin z component by 1.

S+=X

j

[σx(j) +iσy(j)], (3.12) where σ_x and σ_y are the common spin matrices andj enumerates the electron they operate on. Applying theS₊operator once and twice, we get the remaining two states of theS=1 triplett as [26]

30Ψⁿ_m = |Ψ₁Ψ¯₁· · ·Ψ_gΨ¯_gΨ_mΨ¯_n| − |Ψ₁Ψ¯₁· · ·Ψ_gΨ¯_gΨ_nΨ¯_m|

31Ψⁿ_m = |Ψ₁Ψ¯₁· · ·Ψ_gΨ¯_gΨ_mΨ¯_n| (3.13) With a set of such restricted spin orbital functions, which are now insured to be eigenfunctions of the total spin S, Hartree-Fock equations can be derived by variation of the energy with respect to the set of spin orbital wavefunctions, analog to the method described for the closed shell case in section 3.2.1.

Complete active space self-consistent field (CAS-SCF)

The CAS-SCF algorithm uses a set of Slater determinants in the molecular

“active space”, spin orbitals representing molecular states with non-zero total spin (compare level scheme in figure 3.2). Active space orbitals are created by spin ladder operatorS+=P

j[σx(j) +iσy(j)] acting on the Slater determinant of the singly occupied spin orbitals. (The sum runs over the electrons and the σ_x,y symbolise the common spin matrices.) These active orbitals can be thought of as filled with an unpaired electron raised from a paired lower orbital and undergoing a spin flip. This results in a non-zero spin. These orbitals are important for electronic interaction within the supermolecule and have not been accounted for in previously published work. In our numerical work we

3.2. METHODS OF QUANTUM CHEMISTRY 55 have used an active space of four electrons and four orbitals. This spans a space of possible configurations (Slater determinants) for singlet, triplet and quintet (total spinS = 0,1,2) states of O₂, with a single realisation of S = 2, 12 possible realisations ofS = 1 and 35 realisations ofS = 0.

The SCF method to iteratively solve this problem for each of the three total spin configurations (singlet, triplet, quintet) uses a linear combination of Slater determinants ΨI of the active space configurations.

Ψ^Stotal=X

I

c^S_I^totalΨI, (3.14)

wherec_I represents the Slater determinant coupling coefficients, which are pre-determined by spin and spatial symmetry. The introduction of several Slater determinants makes the CAS-ROHF method computationally more difficult than the closed shell Hartree-Fock method.

Figure 3.2: Spin orbital level scheme used for CAS-SCF numerical solution of the O₂-O₂ supermolecule electronic problem using the Gamess quantum chem-istry program. The spin orbitals are classified into three subspaces. Spin or-bitals bracketed inD are doubly occupied with electrons of anti-parallel spins.

Bracket A marks the “active space”, which is only partially occupied, allow-ing multi-configuration non-zero total spin eigenfunctions within a ROHF ap-proach. The V bracket, which is unbounded towards higher orbitals, marks virtually occupied orbitals used in perturbative calculations.

3.2.3 2nd order Møller-Plesset perturbation theory (MP2) As we have outlined, the Hartree-Fock methods presented in the previous sec-tions impose several limitasec-tions on the resulting orbitals. The Hartree-Fock result EHF can be considered to be a 0th-order approximation of the exact electronic energy E_exact. The difference is due to the LCAO approximation ansatz of Hartree-Fock theory. The true combined wavefunction of two or more electrons is not exactly equal to the LCAO anti-symmetrised product of the sin-gle particle wavefunctions used therein. Hartree-Fock theory only accounts for electronic interaction using the average field by integrating over the electronic probability distributions. Thus electrons in the SCF spin orbitals are effectively

too close together and the Hartree-Fock methods yield an overestimate of the true electron-electron interaction.

Improvements to the approximations can be made using perturbation theory approaches, for example. This allows the errors, due to neglected electronic correlations within the Hartree-Fock method, to be reduced by determination of the correlation energy E_c to a higher order.

Ec=Eexact−E_HF (3.15)

Møller-Plesset perturbation theory (MPPT) is an application of Rayleigh-Schr¨odinger many-body perturbation theory (RSPT) using the Hartree-Fock Hamiltonian as a 0th-order HamiltonianH₀ =H_HF. The Hamiltonian used in the perturbation calculations is H=H₀+H_pert, with

H_pert= 1 where J is the Coulomb integral operator and K is the exchange integral op-erator as defined by equations (3.7) and (3.8). This perturbation Hamiltonian

“undoes” the simplifications of Hartree-Fock theory and replaces the electronic interaction with the exactr_ij⁻¹operator. With this Hamiltonian, numerical MP2 implementations are able to recover up to 98% of the correlation energy E_c in most situations [81]. MP2 employs Taylor series expansions of H₀ eigenstates and energies, taking into consideration unoccupied virtual spin orbitals marked in bracket V in the level scheme shown in figure 3.2. Figuratively, correlated electrons avoid each other by populating unoccupiedV orbitals, which are part of the approximative expansion of the exact electronic orbitals, with a non-zero probability.

The Hartree-Fock resultE_HF consists of the 0th and first order energies as obtained from the perturbation Hamiltonian (3.16). E_HF =E⁽⁰⁾+E⁽¹⁾, due to the two similar terms in the respective Hamiltonian H_HF. The first perturba-tion result improving the Hartree-Fock result is thus the second order energy E⁽²⁾. Higher order corrections can also be calculated at increasing numerical expense. For a detailed derivation of MP2 method and algorithm, see [77].

3.3 Ab Initio Computation of an O

− O

Potential Energy Surface

An important input for the simulation of evaporative cooling are the elastic (spin-preserving) and inelastic (spin-changing) scattering rates. A comparison of elastic collision and loss rates, integrating scattering rates over all elastic and inelastic channels, allows judgement about whether evaporative cooling of a molecule will be feasible or not. While these rates are now well-known for alkali atoms, where the potential energy surfaces (PES) are simple and isotropic, the rates are not easily accessible for molecules.

Recently, Avdeenkov and Bohn have calculated oxygen-oxygen scattering rates in [31], based on potential energy surfaces obtained by Bussery and

3.3. AB INITIOCOMPUTATION OF AN O₂−O₂ PES 57