Four-Component Perturbation and Response Theory

The achievements in the first-principles prediction of molecular properties are very impressive and the state of the art is under constant review (see, e.g.. 

Any electromagnetic perturbation of the set of N electrons in a molecule finally leads to sums of operators that may be classified according to the number of electronic coordinates involved as one- or two-electron operators. Of course, if an electron experiences a truly external field a one-electron operator will describe this interaction. If two electrons interact via retarded magnetic fields of the moving electrons as expressed in the frequency-independent Breit interaction and, hence, in the Breit-Pauli Hamiltonian, two-electron interaction operators arise. 

For the sake of simplicity, we may consider one-electron operators and incorporate them in the many-electron Hamiltonian via 

If Af contained two-electron operators, the subsequent analysis given for the one-electron pertubations would be analogous. However, two-electron effects necessarily refer to the description of the electron-electron interaction, while all external potentials affect the potential energy of each electron separately and hence take the form of one-electron operators X(f). Note that these one-electron perturbation operators X must not be confused with the X-operator of decoupling schemes introduced in chapter 11. 

The reason why X can be considered as a simple additive operator is the fact that external vector and scalar potentials hidden in X(f) are simply added to the field-free one-electron Dirac Hamiltonian by the principle of minimal coupling discussed in section 5.4. Therefore, they can easily be separated from the field-free many-electron Hamiltonian Hgi discussed so far. 

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The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

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