Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations

Abstract

Various kinetic balances for constructing appropriate basis sets in four-component relativistic calculations are examined in great detail. These include the well-known restricted (RKB) and unrestricted (UKB) kinetic balances, the less-known dual kinetic balance (DKB) as well as the unknown inverse kinetic balance (IKB). The RKB and IKB are complementary to each other: The former is good for positive-energy states, whereas the latter good for negative-energy states. The DKB combines the good of both RKB and IKB and even provides full variational safety. However, such an advantage is largely offset by its complicated nature. The UKB does not offer any particular advantages as well. Overall, the RKB is the simplest ansatz. Although the negative-energy states by a finite RKB basis are in error of O(c ⁰), there is no objection to using them as intermediates for a sum-over-states formulation of perturbation theory, provided that the magnetic balance is also incorporated in the case of magnetic properties. In particular, the RKB is also an essential ingredient for formulating two-component relativistic theories, while all the others are simply incompatible. As such, the RKB should be regarded as the cornerstone of relativistic electronic structure calculations.

Appendix: The radial Dirac equation for a hydrogenic ion in a DKB Gaussian basis

The radial parts of the DKB basis functions (35) and (36) can be rewritten as

$$ R^L_{\mu}(r)=\frac{1}{r}\pi^L_{\mu}(r),\quad R^S_{\mu}(r)=\frac{1}{r}\pi^S_{\mu}(r), \quad \mu=1,\ldots, N, $$

(68)

such that the radial parts $(P_{n\kappa}(r),Q_{n\kappa}(r))^T=(\frac{1}{r}P^{\prime}_{n\kappa}(r),\frac{1}{r}Q^\prime_{n\kappa}(r))^T$ of spinor ψ _i (34) can be expanded as

$$ \left( \begin{array}{cc} P^\prime_{n\kappa}(r) \\ Q^\prime_{n\kappa}(r) \end{array}\right) = \sum_{\mu=1}^{N} \left( \begin{array}{cc} \pi^L_{\mu} \\ \frac{1}{2mc}\left( \frac{d}{dr}+\frac{\kappa}{r} \right)\pi^L_{\mu} \end{array}\right) A^D_{\mu, n\kappa} +\sum_{\mu=1}^{N}\left( \begin{array}{cc} \frac{1}{2mc}\left( \frac{d}{dr}-\frac{\kappa}{r} \right)\pi^S_{\mu} \\ \pi^S_{\mu} \end{array}\right) B^D_{\mu, n\kappa}. $$

(69)

The matrix representation of the radial Dirac equation

$$ \left( \begin{array}{cc} V & c\left(-\frac{d}{dr}+\frac{\kappa}{r}\right) \\ c\left(\frac{d}{dr}+\frac{\kappa}{r}\right) & V-2mc^2 \end{array}\right) \left( \begin{array}{cc} P^\prime_{n\kappa}(r) \\ Q^\prime_{n\kappa}(r) \end{array}\right) =\left( \begin{array}{cc} P^\prime_{n\kappa}(r) \\ Q^\prime_{n\kappa}(r) \end{array}\right) {\mathbf{\epsilon}}^D $$

(70)

is then of the same form as Eq. 37 but with the matrix elements defined as

$$ {\mathbf{V}}_{\mu\nu}^{LL} =\langle \pi^L_\mu |V|\pi^L_\nu\rangle, $$

(71)

$$ {\mathbf{V}}_{\mu\nu}^{SS} = \langle \pi^S_\mu |V|\pi^S_\nu\rangle, $$

(72)

$$ {\mathbf{S}}_{\mu\nu}^{LL}= \langle \pi^L_\mu |\pi^L_\nu\rangle, $$

(73)

$$ {\mathbf{S}}_{\mu\nu}^{SS}= \left\langle{ \pi^S_\mu |\pi^S_\nu}\right\rangle, $$

(74)

$$ {\mathbf{T}}_{\mu\nu}^{LL}= \tfrac{1}{2m}\left\langle{ \left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) \pi^L_\mu| \left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right)|\pi^L_\nu}\right\rangle =\tfrac{1}{2m}\left\langle {\pi^L_\mu| \left(-\tfrac{d^2}{dr^2}+\tfrac{\kappa(\kappa+1)}{r^2}\right) |\pi^L_\nu}\right\rangle , $$

(75)

$$ {\mathbf{T}}_{\mu\nu}^{SS}= \tfrac{1}{2m}\left\langle{ \left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\mu| \left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right)|\pi^S_\nu}\right\rangle =\tfrac{1}{2m}\left\langle {\pi^S_\mu| \left(-\tfrac{d^2}{dr^2}+\tfrac{\kappa(\kappa-1)}{r^2}\right) |\pi^S_\nu}\right\rangle, $$

(76)

$$ {\mathbf{W}}_{\mu\nu}^{LL}= \left\langle{ \left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) \pi^L_\mu|V \left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right)|\pi^L_\nu}\right\rangle, $$

(77)

$$ {\mathbf{W}}_{\mu\nu}^{SS}= \left\langle {\left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\mu|V \left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right)|\pi^S_\nu}\right\rangle, $$

(78)

$$ \begin{aligned} {\mathbf{W}}_{\mu\nu}^{LS} & = \tfrac{1}{2mc}\left[\left\langle {\pi^L_\mu| V|\left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\nu}\right\rangle +\left\langle {\left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) \pi^L_\mu|V |\pi^S_\nu}\right\rangle\right.\\ &\left.+\tfrac{1}{2m}\left\langle {\left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) \pi^L_\mu| \left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) | \left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\nu}\right\rangle\right], \end{aligned} $$

(79)

$$\begin{aligned}{\mathbf{W}}_{\mu\nu}^{SL} & =\tfrac{1}{2mc}\left[\left\langle {\pi^S_\mu|V|\left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right)\pi^L_\nu}\right\rangle +\left\langle{\left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\mu|V|\pi^L_\nu}\right\rangle\right.\\ & \left.+\tfrac{1}{2m}\left\langle{\left(\tfrac{d}{dr}-\tfrac{\kappa}{r}\right) \pi^S_\mu|\left(-\tfrac{d}{dr}+\tfrac{\kappa}{r}\right) |\left(\tfrac{d}{dr}+\tfrac{\kappa}{r}\right)\pi^L_\nu}\right\rangle \right].\end{aligned}$$

(80)

Note that the element for the above integrals is dr instead of r ² dr. For Gaussian-type functions (52), i.e.,

$$ \begin{aligned} \pi^L_\mu &= r^{n} e^{-\zeta_\mu r^2},\quad n\ge 1, \\ \pi^S_\nu &= r^{n^\prime} e^{-\zeta_\nu r^2},\quad n^\prime\ge 1, \end{aligned} $$

(81)

the integrals are to be evaluated as

$$ {\mathbf{V}}^{LL}_{\mu\nu} =-Z {\mathcal{I}}(2n-1,\zeta_{\mu\nu}),\quad \zeta_{\mu\nu}\doteq \zeta_{\mu}+\zeta_{\nu}, $$

(82)

$$ {\mathbf{V}}^{SS}_{\mu\nu} =-Z {\mathcal{I}}(2n^\prime-1,\zeta_{\mu\nu}), $$

(83)

$$ {\mathbf{S}}^{LL}_{\mu\nu} = {\mathcal{I}}(2n,\zeta_{\mu\nu}), $$

(84)

$$ {\mathbf{S}}^{SS}_{\mu\nu} = {\mathcal{I}}(2n^\prime,\zeta_{\mu\nu}), $$

(85)

$$ \begin{aligned} {\mathbf{T}}^{LL}_{\mu\nu} &=\tfrac{1}{2n^\prime}[4\zeta_\mu\zeta_\nu {\mathcal{I}}(2n+2,\zeta_{\mu\nu}) -2(n+\kappa)\zeta_{\mu\nu}{\mathcal{I}}(2n,\zeta_{\mu\nu}) \\ &\quad+(n+\kappa)^2 {\mathcal{I}}(2n-2,\zeta_{\mu\nu})], \end{aligned} $$

(86)

$$ \begin{aligned} {\mathbf{T}}^{SS}_{\mu\nu} &=\tfrac{1}{2n^\prime}[4\zeta_\mu\zeta_\nu {\mathcal{I}}(2n^\prime+2,\zeta_{\mu\nu}) -2(m-\kappa)\zeta_{\mu\nu}{\mathcal{I}}(2n^\prime,\zeta_{\mu\nu}) \\ &\quad+(n^\prime-\kappa)^2 {\mathcal{I}}(2n^\prime-2,\zeta_{\mu\nu})], \end{aligned} $$

(87)

$$ \begin{aligned} {\mathbf{W}}^{LL}_{\mu\nu} &=-Z[4\zeta_\mu\zeta_\nu {\mathcal{I}}(2n+1,\zeta_{\mu\nu}) -2(n+\kappa)\zeta_{\mu\nu}{\mathcal{I}}(2n-1,\zeta_{\mu\nu}) \\ &\quad+(n+\kappa)^2 {\mathcal{I}}(2n-3,\zeta_{\mu\nu})], \end{aligned} $$

(88)

$$ \begin{aligned} {\mathbf{W}}^{SS}_{\mu\nu} &=-Z[4\zeta_\mu\zeta_\nu {\mathcal{I}}(2n^\prime+1,\zeta_{\mu\nu}) -2(n^\prime-\kappa)\zeta_{\mu\nu}{\mathcal{I}}(2n^\prime-1,\zeta_{\mu\nu}) \\ &\quad+(n^\prime-\kappa)^2 {\mathcal{I}}(2n^\prime-3,\zeta_{\mu\nu})], \end{aligned} $$

(89)

$$ \begin{aligned} {\mathbf{W}}^{LS}_{\mu\nu} &= -\tfrac{Z}{2mc}[ -2\zeta_{\mu\nu}{\mathcal{I}}(n+n^\prime,\zeta_{\mu\nu}) +(n+n^\prime){\mathcal{I}}(n+n^\prime-2,\zeta_{\mu\nu})] \\ &\quad -\tfrac{1}{4m^2c} [ 8\zeta_\mu\zeta_\nu^2{\mathcal{I}}(n+n^\prime+3,\zeta_{\mu\nu}) \\ &\quad-4((n+\kappa)\zeta_\nu^2+(2n^\prime+1)\zeta_\mu\zeta_\nu){\mathcal{I}}(n+n^\prime+1,\zeta_{\mu\nu}) \\ &\quad+2((n^\prime-1+\kappa)(n^\prime-\kappa)\zeta_\mu +(n+\kappa)(2n^\prime+1)\zeta_\nu){\mathcal{I}}(n+n^\prime-1,\zeta_{\mu\nu})\\ &\quad-(n+\kappa)(n^\prime-1+\kappa)(n^\prime-\kappa){\mathcal{I}}(n+n^\prime-3,\zeta_{\mu\nu})]\\ &={\mathbf{W}}^{SL\dagger}, \end{aligned} $$

(90)

where

$$ {\mathcal{I}}(n,\zeta_{\mu\nu})=\int\limits_{0}^{\infty} r^n e^{-\zeta_{\mu\nu} r^2} dr =f_n \frac{(n-1)!!}{(2\zeta_{\mu\nu})^{(n+1)/2}},\quad f_n=\left\{\begin{array}{ll} \sqrt{\pi} & n\in \hbox{non-negative even}, \\ 1 & n\in \hbox{non-negative odd}. \end{array}\right. $$

(91)

Note that DKB also involves new integrals not present in other schemes when evaluating the NMR shielding constants. However, they are not to be presented here.