Last hour: Electron Spin Triplet electrons “avoid each other”, the WF of the system goes to zero if the two electrons approach each other. Consequence: less Coulomb repulsion in triplet states than in singlet states of the same electron configuration. In multielectron systems, the WFs represented by Slater determinants: –N electrons N x N determinant –Each column contains one of the possible spin-orbitals (e.g. 1s ) –Each row contains these spin-orbitals with the “index” of one electron, ensuring antisymmetry w.r. to exchange of two electrons. –Normalization factor:
2+ e1e2 Hartree Self Consistent Field Method actual situation: electrons with correlated motion
Basic considerations about multielectron atoms: Perturbation theory problematic, because H’ is a large perturbation either we need high orders of P.T., or the result will be very inaccurate. Variational theory better suited. In principle, however, we have to change the functional form of the trial w.f., not only parameters linear variation functions. To have a physical picture of the result, we choose the trial w.f.’s to be products of 1-e - -w.f.’s. This is not accurate, as H’ is not separable, so the “true” w.f. cannot be written as a product of n 1-e - -w.f.’s !
2+ e1e2 Start e.g. with H functions Sort electrons into orbitals, two for each orbital (Pauli Exclusion Principle!) Result: starting “population” of orbitals with electrons Hartree Self Consistent Field Method
Hartree Self Consistent Field (SCF) Approach: Choose a trial wave function 0 with where each s j is a product of a normalized function of r j and a spherical harmonic: s j (r j, j, j ) = h(r j )·Y ℓ m ( j, j ) Assume that we can treat each electron as moving through an average charge distribution caused by the other electrons. Focus on potential energy term for electron j: e-e-repulsion averaged over e - -cloud
2+ e1 approximation: assume that all electrons except one are “smeared out” only treat average field from electron population of orbitals solve radial Schrödinger equation to get new shape of orbital Hartree Self Consistent Field Method
2+ e2 Hartree Self Consistent Field Method repeat this process, “focusing” once on each electron result: “new & improved” radial functions for each electron
Central field approximation: Assume that the actual electric field only depends on r: Note: 1 st approximation: WF=product of one-e - fcts (only true for separable Hamiltonian) 2 nd approximation: treat each electron in the average field of all other electrons (clearly an approximation!) 3 rd approximation: assume central fields (also clearly an approximation!) although s j looks like a H-like w.f., the field is not a simple Coulomb potential, so h(r j ) is not a H-like radial function! maintains idea of orbitals for each electron
How to get the wave functions: Recall that, in principle, we have to change functional form very impractical for systematic automated minimization procedure Expand radial functions in full basis set:h j = k a jk k where k are basis functions. Minimize energy by variation of the a j Slater-type radial functions (STOs) are sometimes used as basis functions: normalization leading term in Laguerre pol. = (Z-s)/n is called “orbital exponent”. Determines “size” of radial function The parameter s is a “screening constant” that reduces the effective nuclear charge. Another possibility (very common): Gaussian functions mimicking STO’s Usually, several STO’s or other orbitals needed for each radial function to get the correct number of nodes, etc.
2+ e1 Calculate total energy Repeat previous step with “new & improved” functions until no significant changes from one iteration to the next. Hartree Self Consistent Field Method
Strategy: Iterative procedure 1.Choose “reasonable” starting wave functions: H-like w.f.’s 2.“Fill” the orbitals paying attention to the Pauli Exclusion Principle, never putting more than two electrons in the same orbital. 3.Use w.f.’s for electrons to approximate V j (r j ) makes H^ separable into H j and H k j 4.Use V j to solve the radial Schrödinger equation 5.for improved w.f.’s h j, determine energies of orbitals E j 6.Return to (3) with improved w.f.’s h j. Repeat until “no significant” change in s j , and E j after each new iteration (“self-consistent”).
Note: E j represents energy of electron j in the j-th orbital. Each E j contains the repulsion with each other electron, e.g., electron k. To avoid “double-counting” of repulsion energies, the total energy is given by Radial w.f.’s of Hartree-orbitals are not H-like radial functions, but the solutions are still labeled by quantum numbers n, ℓ. Angular w.f.’s are the same Y ℓ m as in H-atom (by design!) Set of orbitals with the same n shell Set of orbitals with the same ℓ subshell Filled shells and subshells yield a spherically symmetric probability density (Unsöld’s Theorem)
Hartree-Fock SCF Method Need to include spin explicitly use Slater determinants as total wave function instead of just the product HF-SCF (“configuration”) yields approximate wave function for n-e - atom HF-SCF ground state energy: Recall ground state from degenerate perturbation theory for He: where (Coulomb integral) and (exchange integral) The effective Hamiltonian of electron j for the HF-SCF method can be written as
where J k and K jk are the operators similar to the Coulomb and exchange integrals, F is called the Fock operator. (details see e.g. Szabo and Ostlund “Modern Quantum Chemistry”) In practice: For N electrons, use M spatial orbitals as basis functions, yields 2M spin-orbitals (M up, M down), 2M ≥ N N lowest spin-orbitals are called “occupied” 2M – N remaining spin-orbitals are called “virtual” or “unoccupied” larger and more complete basis set lower energies converges to “Hartree-Fock limit”
Summary: Despite many approximations in HF-SCF, the w.f.’s and energies obtained are reasonably good. Example:He(1s 2 ) E HF = eV whileE exact = eV Note that E HF > E exact (linear variation method!) HF-SCF gives us the basis for the buildup principle that determines the ordering of low energy configurations for multi-electron atoms Concept of as product of 1-e - atomic orbitals (AO’s) is approximate, but our only chance to discuss multi-electron systems in a simple way. The energies of the 1-e - AO’s correspond approximately to the negative of the ionization energies of these electron, i.e., j is about the energy necessary to remove an electron from the orbital j (Koopmans’ Theorem).
Comparison of Koopmans’ Theorem with exp. Ionization Energies from McQuarrie & Simon: Physical Chemistry