Download Variational Theory - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 8.2: An aside: Variational Theory, another approximation method = an approximation method which works even when perturbation theory fails. Especially useful and important for many-electron systems. oEHE ≥= ψψ ~|ˆ|~~Theorem: oE E ψ~ ~ Approximate energy of the system “any” normalized function of the coordinates of the system including spin. = trial wave function which must satisfy the usual QM boundary conditions. Lowest energy eigenvalue for the system (ground state) docsity.com Proof: Derivation ahead oψψ = ~ (1) Set = exact ground state wave function. oooooo EEHE ===⇒ ψψψψ ||ˆ| ~ oψψ = ~ exactlyi.e. the theorem holds for oψψ ≠ ~ which is usually the case(2) When Will use our theorem that any arbitrary function can be expressed as an expansion in a complete basis set. Use expansion postulate to expand ψ~ in terms of a complete set of eigenfunctions of H. ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ≤≤∞= == =⇒ ∑ ∞ = ...;,,2,1,0 |;ˆ~ 210 EEEk EH a o kkkkk k kk K ll δψψψψψψ docsity.com This theorem is the basis of the variational method for calculating approximate wave functions and energies of many electron atoms and molecules Note: The theorem holds for the lowest energy state for each state of a given symmetry, if ψ~ is chosen to have the correct symmetry; for example, the lowest s state, p state, d state etc. l, ~ symEE ≥⇒ docsity.com Variational Method ψψ ~~~ˆ EH = * Start with ~ψ integrate, and rearrange for EPremultiply by )1(~|~ ~|ˆ|~ ψψ ψψ H E =⇒ i N i i fc∑ = = 1 ~ψLet: where {fi} are functions that satisfy the general conditions for a wave function. They may be constitute an orthonormal set or they may not. docsity.com )2( | |ˆ| ~ 1 1 * 1 1 * 1 1 * 1 1 * ∑∑ ∑∑ ∑∑ ∑∑ = = = = = = = = == N j N i jiij N j N i jiij N j N i ijij N j N i ijij Scc Hcc ffcc fHfcc E where ijjiijji ffSfHfH |;|ˆ| == Hji is a Hamiltonian matrix element and Sji is an overlap integral Rearrange (2) such that )3(~ 1 1 * 1 1 * ∑∑∑∑ = == = = N j N i jiij N j N i jiij HccSccE docsity.com For N basis functions, the secular equation is given by: 0 2211 2222222121 1112121111 = −−− −−− −−− NNNNNNNN NN NN ESHESHESH ESHESHESH ESHESHESH K MMMM K K The secular equation yields a Nth order polynomial in E. For each E can solve for {ci} to get the “best” approximate wave function for the system having that E. docsity.com Back to the problem: 8.3: Optimization of the Radial Part by the Variation Method What function should we use for the individual entries, φi(k) in the determinant? Instead of the H-atom eigenfunctions, we use modified functions. The key difference in the modified functions from the H-atom functions Effective nuclear charge Zeta (ζ) < the true nuclear charge The outermost electrons are shielded from the full nuclear charge by other electrons. Each 1-e- orbital is constructed from a linear combination of H-atom-like orbitals. All coefficients in the linear combination in each orbital are used as variational parameters. The ζ values are optimized separately. docsity.com Next: solve the Schrodinger Equation for each electron i. To do so, we must know Vieff . This means that we must know the functional forms of all the other orbitals, also the case for the remaining n-1 electrons. That is, the answers must be known to solve the problem. The way out of this quandary is to use an iterative approach. ● Make a reasonable guess for an initial set of orbitals ● Calculate an effective potential using these orbitals ● Calculate the energy and orbital functions for each of the n electrons in turn ● New orbitals are used to refine the initial guesses Repeat until the solutions for the energies and the orbitals are self-consistent docsity.com
