Transformed Hamiltonian introduced approximations

The NpPolMe basis sets were developed recently (10) for the investigation of relativistic effects using the DK transformed hamiltonian (13, 18-20). This is the spin-averaged no-pair approximation which reduces the 4-component relativistic one-electron hamiltonian to a 1-component form without introducing strongly singular operators. NpPolMe basis sets indirectly incorporate some relativistic effects on the wave function. Let us note that both PolMe and NpPolMe contracted sets share the same exponents of primitive Gaussians. Contraction coefficients are, however,... 

The approximation we make in the transformation process is to neglect both the off-diagonal terms in the transformed Hamiltonian and the residual coupling in the wave function. At the same time as neglecting 8, we also renormalize rlf. This will introduce a scaling factor, which we will absorb into the definition of and O. With these approximations, the relations between the wave functions become... 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

This procedure is usually known as the elimination of the small component (ESC), and Eq. (34) is still equivalent to the original Dirac equation. Although the equation has been reduced to a two-component form, nothing is gained since we now have an energy-dependent Hamiltonian, and one must introduce further approximations to transform Eq. (34) into a form useful for actual calculations. The principal difference between the Pauli and the ZORA Hamiltonian is that to obtain the Pauli Hamiltonian, one uses an expansion in c ... 

Note that no approximation has been made so far. The Breit-Pauli (BP) approximation [49] is introduced by expanding the inverse operators in the Schrodinger-Pauli equation in powers of (V — E) jlc and ignoring the higher-order terms. Instead, the BP approximation can be obtained truncating the Taylor expansion of the FW transformed Dirac Hamiltonian up to the (ptcf term. The one-electron BP Hamiltonian for the Coulomb potential V = Zr /r is represented by... 

The partition function Eq. 6 describes a system of mutually interacting chains. Introducing auxiliary fields, U and W, via a Hubbard-Stratonovich transformation, one can decouple the interaction between the chains and rewrite the Hamiltonian in terms of independent chains in fluctuating fields. Then, one can integrate over the chain conformations and obtain a Hamiltonian which only depends on the auxiliary fields. Thermodynamic averages like density or structure factors can be expressed as averages over the fields, U and W, without approximation. 

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