“Brillouin's theorem”的意思、由来-开放百科全书

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Brillouin's theorem

释义

Proof
References
Further reading

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, states that given a self-consistent optimized Hartree-Fock wavefunction , the matrix element of the Hamiltonian between the ground state and a single excited determinant (i.e. one where an occupied orbital a is replaced by a virtual orbital r) must be zero.

This theorem is important in constructing a configuration interaction method, among other applications.

Another interpretation of the theorem is that the ground electronic states solved by one-particle methods (such as HF or DFT) already imply configuration interaction of the ground-state configuration with the singly excited ones. That renders their further inclusion into the CI expansion redundant.^[1]

Proof

The electronic Hamiltonian of the system can be divided into two parts: one consisting of one-electron operators and the other of two-electron operators . In methods of wavefunction-based quantum chemistry which include the electron correlation into the model, the wavefunction is expressed as a sum of series consisting of different Slater determinants (i.e., a linear combination of such determinants). In the simplest case of configuration interaction (as well as in other single-reference multielectron-basis set methods, like MPn, etc.), all the determinants contain the same one-electron functions, or orbitals, and differ just by occupation of these orbitals by electrons. The source of these orbitals is the converged Hartree–Fock calculation, which gives the so-called reference determinant with all the electrons occupying energetically lowest states among the available. All other determinants are then made by formally "exciting" the reference one (one or more electrons are cleared from one-electron states occupied in and put into states unoccupied in ). As the orbitals remain the same, we can simply transition from the many-electron state basis (, , , …) to the one-electron state basis (which was used for Hartree–Fock: , , , , …), greatly improving the efficiency of calculations. For this transition, we apply the Slater–Condon rules and evaluate

which we recognize is simply an off-diagonal element of the Fock matrix . But the reference wave function was obtained by the Hartree–Fock calculation, or the SCF procedure, whole point of which was to diagonalize the Fock matrix. Hence for an optimized wavefunction this off-diagonal element must be zero.

This can be made evident also if we multiply both sides of a Hartree–Fock equation

by and integrate over the electronic coordinate:

.

As the Fock matrix has already been diagonalized, the states and are the eigenstates of the Fock operator, and as such are orthogonal; thus their overlap is zero. It makes all the right-hand side of the equation zero:^[1]

,

which proves the Brillouin's theorem.

The theorem have also been proven directly from the variational principle (by Mayer) and is essentially equivalent to the Hartree–Fock equations in general.^[2]

References

1. ^¹{{cite book |last1=Tsuneda |first1=Takao |title=Density Functional Theory in Quantum Chemistry |date=2014 |publisher=Springer |location=Tokyo |isbn=978-4-431-54825-6 |pages=73–75 |chapter=Ch. 3: Electron Correlation|doi=10.1007/978-4-431-54825-6 }}
2. ^{{cite book |last1=Surján |first1=Péter R. |title=Second Quantized Approach to Quantum Chemistry |date=1989 |publisher=Springer |location=Berlin, Heidelberg |isbn=978-3-642-74755-7 |pages=87–92 |chapter=Ch. 11: The Brillouin Theorem|doi=10.1007/978-3-642-74755-7_11 }}

Proof

References

Further reading