Lecture 3: Quantum Satis*

Transcription

1 Lecture 3: Quantum Satis* Last remarks about many-electron quantum mechanics. Everything re-quantized! * As much as needed, enough.

2 Electron correlation Pauli principle Fermi correlation Correlation energy E corr = E exact E HF Dynamical correlation = inability of the single-sd wave function to model the correlation hole and dispersion interaction Static correlation = near-degeneracies of HF molecular orbitals No true difference But see for another opinion e.g. Hollett & Gill, J. Chem. Phys. 134, (2011)

3 Variation principle Variation principle = solution of the eigenvalue equation is equivalent to optimization of the energy functional By expressing the electronic state in terms of a finite set of numerical parameters, the stationary points of the function are the approximate states with energy E(C)

4 Optimization of approximate wave functions Approximate wave function In order to find the stationary points of the energy function, we form the first and second derivative of it: We can resort to a first or a second-order direct optimization scheme for finding the parameters C

5 Optimization of the approximate wave function Or noting that the gradient becomes zero when That (obviously) corresponds to the eigenvalue problem HC = E(C)C, where The m solutions of this problem correspond to the approximate wave function with corresponding approximate wave functions The eigenvector corresponding to the lowest eigenvalue is the minimum, the second a first-order saddle point, and so on.

6 Final remarks about electron densities Recall these: P p Ο Now the energy of any state of any system can be expressed as

7 Final remarks about electron densities In fact, you do not need even the two-electron density matrix Hohenberg & Kohn (1964): The ground state density uniquely determines the potential and thus all properties of the system, including the many-body wave function In particular, the functional F[r]=T[r]+U[r] is a universal functional of the density For any potential the density functional E[] r < F[] r Úv()() r r r d r obtains its minimal value at the ground-state electron density in the potential v(r)

8 So far: observables operators states functions Second quantization The formalism of second quantization: also states are represented as operators Allows us to concentrate on the few matrix elements of interest, thus avoiding the need for dealing directly with the many-particle wave function and the coordinates of all the remaining particles

9 Occupation number vectors In SQ, each Slater determinant is represented as an occupation number vector There is one-to-one mapping between ON s and SD s, but ON s have no space-like reference Inner product Sc. vacuum state

10 Creation and annihilation operators In SQ, all operators are constructed as sequences of elementary creation and annihilation operators Creation operator puts an electron into an unoccupied orbital Annihilation operator removes an electron from an occupied orbital

11 Creation and annihilation operators Important anticommutation relations

12 Operators First quantization One-electron operator Second quantization One-electron operator Two-electron operator Operators independent on the spin-orbital basis Operators dependent on the number of electrons Exact operators Two-electron operator Operators depend on the spin-orbital basis Operators independent on the number of electrons Projected operators

13 Operators For example, the molecular Hamiltonian is in the SQ formalism  H < h aa g aaaa PQ  PQ P Q PQRS P R Q S PQRS Where the amplitudes (matrix elements) are just

14 Commutators and anticommutators When working with SQ formalism, we will need to evaluate commutators and anticommutators consisting of strings of the elementary operators Some identities helpful in simplification of the commutators

15 Spin in second quantization SQ formalism remains unchanged if spin degree of freedom is treated explicitly, e.g. Now operators can be spin-free, mixed or spin operators Spin-free operators depend on the orbitals but have identical amplitudes for alpha and beta spins Spin operators are independent on the functional form of the operators Mixed operators depend on both

16 Spin in second quantization An example of a spin-free operator is the molecular electronic Hamiltonian that can be written as One-electron singlet excitation operator Two-electron singlet excitation operator pqrs p r s q pq rs qr ps e < Âa a aa < E E,dE t u t u tu

17 Things to think about Could we construct a wave-function free approach on electronic structure, basing on generalized density matrices (see section in the lecture notes)? This does not refer to density-functional theory, but a theory having the (exact) two-particle density matrix involved See the publications of David Mazziotti, e.g. Chem. Rev. 112, (2011). Verify the commutation relations (A.39-44) in the lecture notes

18 Homework Study the Section 2.4 and Appendix A Familiarize yourself with the Chapter 3 Complete the preparatory assignment of Exercise 2 Think about the project works Alone or in a group, with whom? Possible topics Feel free to announce it at any time; deadline for the topics is on April 11