Scalar relativistic Hamiltonians

Code author: Prakash Verma and Francesco A. Evangelista

Section author: Prakash Verma, Wallace D. Derricotte, and Francesco A. Evangelista

The exact-two-component (X2C) approach is a convenient way to introduce scalar relativistic effects in DFT and wave function-based methods. PSI4 implements the spin-free one-electron version of X2C, which produces a modified one-electron Hamiltonian $H_{\mathrm{X}2\mathrm{C}}$:

$$H_{\mathrm{X}2\mathrm{C}}=T_{\mathrm{X}2\mathrm{C}}+V_{\mathrm{X}2\mathrm{C}}$$

that is a sum of a kinetic energy ($T_{\mathrm{X}2\mathrm{C}}$) and potential energy ($V_{\mathrm{X}2\mathrm{C}}$) operator. Our implementation is equivalent to the one reported by Cheng and Gauss [Cheng:084114]. X2C calculations require the use of special (alternatively fully uncontracted) basis sets designed for relativistic calculations. Common choices include the Dunning Douglass–Kroll basis sets (cc-pVXZ-DK, cc-pCVXZ-DK, cc-pwCVXZ-DK) and Roos’ ANO basis sets.

Note

See also Interface to DKH by A. Wolf, M. Reiher, and B. A. Hess for another relativistic Hamiltonian.

A First Example

The following is a simple input that will perform a Hartree–Fock calculation using the X2C Hamiltonian.

molecule {
  H
  F 1 0.92
}

set {
    scf_type pk
    basis cc-pvdz
    relativistic x2c
}

energy('hf')

This computation yields the following result:

@RHF Final Energy:  -100.10007984692388

 => Energetics <=

  Nuclear Repulsion Energy =              5.1767335622934780
  One-Electron Energy =                -150.7611816259664579
  Two-Electron Energy =                  45.4843682167491039
  Total Energy =                       -100.1000798469238902

while a non-relativistic calculation yields the following energy:

@RHF Final Energy:  -100.01928891411315

 => Energetics <=

  Nuclear Repulsion Energy =              5.1767335622934780
  One-Electron Energy =                -150.6645256529074572
  Two-Electron Energy =                  45.4685031765008461
  Total Energy =                       -100.0192889141131474

Basis sets options

The X2C module in PSI4 supports different combinations of basis set. By default, if the input file specifies only BASIS, then the X2C module will solve the modified Dirac equation in an uncontracted basis and then recontract the X2C Hamiltonian in the original basis. Alternatively, the user can use BASIS_RELATIVISTIC to specify a different basis set to solve the modified Dirac equation.

set {
    basis cc-pvdz-dk
    basis_relativistic cc-pvtz-dk
    relativistic x2c
}

It is recommended that when employing the X2C relativistic Hamiltonian, that you use a fully decontracted basis set. This can be done simply in the input by adding “-decon” to the name of the primary basis you want to use for the calculation as detailed in Decontracted Basis Sets. Publications resulting from the use of X2C should cite the following publication: [Verma:2015]

Theory

X2C is based on exact decoupling of positive-energy ( $h_{++}^{FW}$ ) and negative-energy ($h_{--}^{FW}$ ) blocks of the Dirac Hamiltonian ($h^D$).

$$\begin{array}{r}{U}^{†}{h}^{\mathrm{D}}U=U^{†}\left(\begin{array}{cc}{h}_{LL}& h_{LS}\\ h_{SL}& h_{SS}\end{array}\right)U=\left(\begin{array}{cc}{h}_{++}^{\mathrm{F}\mathrm{W}}& 0\\ 0& h_{--}^{\mathrm{F}\mathrm{W}}\end{array}\right)\end{array}$$

The transformation ( $U$ ) is obtained from the solutions of the Dirac equation in kinetically balanced basis [Kutzelnigg:1984] treatment. In the X2C treatment, the positive-energy block of the Hamiltonian ( $h_{++}^{FW}$ ) is given by the sum of a transformed kinetic ($T_{\mathrm{X}2\mathrm{C}}$) and potential energy ( $V_{\mathrm{X}2\mathrm{C}}$ ) contribution. Relativistic kinetic energy ( $T_{\mathrm{X}2\mathrm{C}}$ ) and nuclear-electron interaction potential ( $V_{\mathrm{X}2\mathrm{C}}$ ) is given in terms of non-relativisitc kinetic ($T={\hat{p}}^2/2$) energy and nuclear-electron interaction potential ($V$), coupling matrix ( $X$) and renormalization matrix ( $R$).

$$T_{\mathrm{X}2\mathrm{C}}=R^{†}(TX+X^{†}T-X^{†}TX)R$$

$$V_{\mathrm{X}2\mathrm{C}}=R^{†}(V+\frac{1}{4c^2}{X}^{†}{W}^{\text{SF}}X)R$$

The coupling matrix ( $X=C^S(C^L{)}^{-1}$ ) is obtained from the large ($C^{\mathrm{L}}$) and small ($C^{\mathrm{S}}$) components of the $N$ positive energy solutions of the Dirac equation. The renormalization matrix $R=S^{-1/2}(S^{-1/2}\tilde{S}{S}^{-1/2}{)}^{-1/2}{S}^{1/2}$, depends on the modified overlap matrix $\tilde{S}=S+\frac{1}{2c^2}{X}^{†}TX$. The integrals $W_{\mu \nu }^{\mathrm{S}\mathrm{F}}=⟨{\chi }_{\mu }|\hat{p}\cdot (\hat{V}\hat{p})|{\chi }_{\nu }⟩$ can be easily computed as derivatives of the nuclear-electron attraction integrals with respect to nuclear coordinates. Existing nonrelativistic electronic structure code can be extended to include scalar relativistic effects treated with the X2C method by replacing nonrelativistic kinetic and potential energy with the corresponding X2C operators $T_{X2C}$ and $V_{X2C}$. It is important to note that fully uncontracted basis in needed for the construction of X2C Hamiltonian as Foldy-Wouthuysen (FW [FW:1950]) transformation is obtained in kinetically balance basis.

Keywords

RELATIVISTIC

Relativistic Hamiltonian type

• Type: string

• Possible Values: NO, X2C

• Default: NO

BASIS_RELATIVISTIC

Auxiliary basis set for solving Dirac equation in X2C and DKH calculations. Defaults to decontracted orbital basis.

• Type: string

• Default: No Default