THE HARTREE FOCK METHOD
Download as PPTX, PDF6 likes8,727 views
The document discusses the Hartree and Hartree-Fock methods used for approximating the wave functions of many-electron systems, emphasizing major simplifications and the importance of the Pauli exclusion principle in constructing antisymmetric wave functions. It describes various computational techniques, including the self-consistent field approach and the Slater determinant, to derive electronic energy levels and optimize accuracy. Ultimately, the Hartree-Fock method provides approximate wave functions with minimal error in total energy, making it advantageous for theoretical studies in quantum mechanics.
![Remedy
𝐹(𝑖) = ℎ(𝑖) + 𝑗=1
𝑛/2
[2𝐽𝑗 𝑖 − 𝐾𝑖 𝑖 ]](https://image.slidesharecdn.com/thegod-171028040010/85/THE-HARTREE-FOCK-METHOD-15-320.jpg)
THE HARTREE FOCK METHOD
- 3. PERTURBATION THEORY VARIATIONAL PRINCIPLE WKB APPROXIMATION Judge the quality of the wave functions by the energy Any approximation to the ground state wave function will yield an expectation value of the Hamiltonian that is greater than or equal to the ground state energy i.e.
- 4. The Born-Oppenheimer Approximation The motion of atomic nuclei and electrons in a molecule can be separated. ψ 𝑇=ψ 𝑒×ψ 𝑛
- 5. HARTREE APPROXIMATION • The simplest approximation. Major simplifications
- 6. Electronic Schrödinger Equation for a System of Many Electrons 1 2 𝑖=1 𝑁 𝛻2 KINETIC ENERGY OF ELECTRONS INTERACTION BETWEEN ALL NUCLEI AND ELECTRONS 𝒊=𝟏 𝑵 𝒋>𝒊 𝑵 𝟏 𝒓𝒊𝒋 COULOMBIC REPULSION BETWEEN ELECTRONS 𝒊=𝟏 𝑵 𝑨=𝟏 𝑴 𝒁 𝑨 𝒓 𝒂𝒊
- 7. Hartree’s Self-Consistent Field (SCF) Approach. For a trial wave function ,It permits the many-particle problem to be reduced to problem of single particle. This method invokes orbital approximation i.e. Complex wave function of many electron system ψ (r1, r2,…..,𝑟𝑛) = ψ𝑖(𝑟𝑖). WHATIS SCF METHOD? An iterative method
- 9. 𝑯 𝒆𝒍𝒆𝒄= 𝟏 𝑵 𝒉 𝒆𝒍𝒆𝒄 𝒊 Ideal method for a computer as easily written as an algorithm. Gaussian Wave function Calculate Charge Density Calculate Potential Solve Schrodinger equation Calculate Charge density Is Charge Density Same as before? YES NO ST O P
- 10. Comparison with other approximation: Perturbation theory with hydrogen- like wave functions Variational theory with effective Z Numerical Hartree(- Fock) result Experi mental result E=-5.5𝑅 𝑦 E=-5.695Ry E=-5.724Ry E=-5.807Ry
- 11. Hatree-Fock Method • In 1930,Fock pointed out that Hatree wave function violates Pauli exclusion principle. Two identical fermions cannot occupy the same quantum state simultaneously Wave function has to be anti-symmetric.
- 12. • Antisymmetric Function is defined as ψ (r1,r2)=-ψ (r2,r1). • Main simplification : In this model we need to use simple possible antisymmetric space one can imagine : Antisymmetric Tensor product. • For N=2, we have where α is the electron spin Eigen function.
- 13. Slater Determinant • A function which is determinant of atomic orbitals Slater determinant. • For many electrons Wave function of a multi-electron system that satisfies 1.Anti-symmetric requirements 2.The Pauli exclusion principle.
- 14. Schrodinger Energy in ground state for a single slater determinant Hatree-Fock energy. The quantity we can measure is Electronic density ρ. Answer is Energy Determine energy directly from the minimizing orbitals BUT………
- 15. Remedy 𝐹(𝑖) = ℎ(𝑖) + 𝑗=1 𝑛/2 [2𝐽𝑗 𝑖 − 𝐾𝑖 𝑖 ]
- 16. CRUCIAL OBSERVATION • In theory HF limit is achieved by an infinite basis set. • In practice finite basis sets that can approach HF limit as efficiently as possible. • Hatree-Fock energy is invariant under Unitary transformation on ψ 𝟏to ψ 𝑵. • Non-linear problem , so HF ground state energy will be smallest Eigen value and Eigen operator of Fock operator. “Hartree-Fock method, give us approximate wave functions for the atoms which have errors in total enery of only a fraction of 1%” - J. C. SLATER