New Strategies in Modeling Electronic Structures and Properties …
127
The resulting blocks in the transformed Hamiltonian ¯
H D are two-component Hamiltonians and act on electronic and positronic states only. The exact form of the unitary
transformation U is, however, only known for the free-particle Dirac equation and
is called the Foldy–Wouthuysen transformation [40]. An approximate decoupling
scheme for the many-electron Dirac equation in quantum chemistry was proposed
by Hess. The so-called Douglas–Kroll–Hess (DKH) method [38, 61, 123] is based on
the Foldy–Wouthuysen transformation [40] and represents an order-by-order expansion (in the external potential V ), where the electronic and positronic components
of the Dirac equation are separated iteratively. The DKH transformed Hamiltonian
of (n + 1)-th order has the general form
H n+1 = U
†
n U
†
n−1 . . . U
†
2 U
†
1 H 1 U 1 U 2 . . . U n−1 U n ,
(10)
where H 1 is the free-particle Foldy–Wouthuysen (fpFW) transformed Dirac Hamiltonian H 1 = U fpFW
† H D U fpFW . Thus, different orders of approximations are obtained
by applying subsequent unitary transformations to the relativistic Dirac Hamiltonian [123, 124, 157, 165]. Specifically, the second-order Douglas–Kroll–Hess
(DKH2) Hamiltonian is most commonly used in quantum chemistry as it provides
satisfactory results for conventional chemical problems. In DKH2, only one unitary
transformation U 1 has to be applied. We should note that the explicit form of the
unitary transformation U does not affect lower order DKH Hamiltonians and hence
the operators U i can be represented in different ways, using, for instance, a power
series expansion of an (anti-Hermitian) operator.
The exact two-component (X2C) relativistic Hamiltonian is based on exact decoupling of the large and small components of the Dirac Hamiltonian in its matrix
representation. Specifically, the X2C method exploits the non-symmetric Algebraic
Riccati Equation (nARE) [75, 76], a quadratic matrix equation. The nARE approach
was used for the Dirac Hamiltonian for the first time by Ilias and Saue [65] and introduced as the X2C method. Most importantly, the eigenvalues of the X2C Hamiltonian
are identical to the positive energy branch of the four-component Dirac Hamiltonian.
One should stress that in the majority of quantum chemical applications, these
two-component Hamiltonians account only for scalar relativistic effects and thus only
have a one-component form. Due to this one-component nature, such Hamiltonians
can be easily interfaced with standard quantum chemistry codes. Spin-orbit coupling
effects can be included a posteriori using the spin-orbit configuration-interaction
approach, where the relativistic Hamiltonian is diagonalized in the spin-free basis [97,
154]. To further decrease the computational cost, the spin-orbit integrals are often
calculated within the atomic mean-field intergrals (AMFI) approach [62, 98, 130].
The computationally most efficient way of including relativistic effects in the
Schrödinger equation is to introduce scalar relativistic effects using relativistic
effective core potentials (RECP) [37]. Such a crude approximation is usually sufficiently accurate for chemistry as the influence of the core electrons on the valence
shell (that is the shell containing electrons of relevance in chemical processes) is
rather indirect and can be accurately modelled using parametrized effective pseudopotentials in conjunction with scalar relativistic interactions [37]. Besides being
127
The resulting blocks in the transformed Hamiltonian ¯
H D are two-component Hamiltonians and act on electronic and positronic states only. The exact form of the unitary
transformation U is, however, only known for the free-particle Dirac equation and
is called the Foldy–Wouthuysen transformation [40]. An approximate decoupling
scheme for the many-electron Dirac equation in quantum chemistry was proposed
by Hess. The so-called Douglas–Kroll–Hess (DKH) method [38, 61, 123] is based on
the Foldy–Wouthuysen transformation [40] and represents an order-by-order expansion (in the external potential V ), where the electronic and positronic components
of the Dirac equation are separated iteratively. The DKH transformed Hamiltonian
of (n + 1)-th order has the general form
H n+1 = U
†
n U
†
n−1 . . . U
†
2 U
†
1 H 1 U 1 U 2 . . . U n−1 U n ,
(10)
where H 1 is the free-particle Foldy–Wouthuysen (fpFW) transformed Dirac Hamiltonian H 1 = U fpFW
† H D U fpFW . Thus, different orders of approximations are obtained
by applying subsequent unitary transformations to the relativistic Dirac Hamiltonian [123, 124, 157, 165]. Specifically, the second-order Douglas–Kroll–Hess
(DKH2) Hamiltonian is most commonly used in quantum chemistry as it provides
satisfactory results for conventional chemical problems. In DKH2, only one unitary
transformation U 1 has to be applied. We should note that the explicit form of the
unitary transformation U does not affect lower order DKH Hamiltonians and hence
the operators U i can be represented in different ways, using, for instance, a power
series expansion of an (anti-Hermitian) operator.
The exact two-component (X2C) relativistic Hamiltonian is based on exact decoupling of the large and small components of the Dirac Hamiltonian in its matrix
representation. Specifically, the X2C method exploits the non-symmetric Algebraic
Riccati Equation (nARE) [75, 76], a quadratic matrix equation. The nARE approach
was used for the Dirac Hamiltonian for the first time by Ilias and Saue [65] and introduced as the X2C method. Most importantly, the eigenvalues of the X2C Hamiltonian
are identical to the positive energy branch of the four-component Dirac Hamiltonian.
One should stress that in the majority of quantum chemical applications, these
two-component Hamiltonians account only for scalar relativistic effects and thus only
have a one-component form. Due to this one-component nature, such Hamiltonians
can be easily interfaced with standard quantum chemistry codes. Spin-orbit coupling
effects can be included a posteriori using the spin-orbit configuration-interaction
approach, where the relativistic Hamiltonian is diagonalized in the spin-free basis [97,
154]. To further decrease the computational cost, the spin-orbit integrals are often
calculated within the atomic mean-field intergrals (AMFI) approach [62, 98, 130].
The computationally most efficient way of including relativistic effects in the
Schrödinger equation is to introduce scalar relativistic effects using relativistic
effective core potentials (RECP) [37]. Such a crude approximation is usually sufficiently accurate for chemistry as the influence of the core electrons on the valence
shell (that is the shell containing electrons of relevance in chemical processes) is
rather indirect and can be accurately modelled using parametrized effective pseudopotentials in conjunction with scalar relativistic interactions [37]. Besides being