Spin–orbit coupling matrix elements from the explicitly connected expressions of the response functions within the coupled-cluster theory

Figures & data

Table 1. Terms included in the XCC transition moments calculations.

Leading MBPT order	CCSD	CC3
0	All	All
1	All	All
2	All	Neglect $〈{\mu }_3|\cdots |{\nu }_3〉$
3	All	Neglect $〈{\mu }_3|\cdots |{\nu }_3〉$
		Neglect $〈{\mu }_2|\cdots |{\nu }_3〉$ unless includes $T_1$ or $S_1$
		Neglect $〈{\mu }_1|\cdots |{\nu }_3〉$ unless includes $T_1$ or $S_1$

Table

Symbol	Meaning
$\mathcal{T}(N-M)$	Reduced transition moment from state N to M
$\mathcal{A}(N-M)$	Transition probability from state N to M
$\mathbf{\text{d}}=(d_{-1},d_0,d_1)$	Dipole moment operator
$\mathbf{\text{Q}}=(Q_{-2},Q_{-1},Q_0,Q_1,Q_2)$	Quadrupole moment operator
XCCSD(G)	This work, CCSD approximation, Gaussian basis set
XCC3(G)	This work, CC3 approximation, Gaussian basis set
XCCSD(S)	This work, CCSD approximation, Slater basis set
XCC3(S)	This work, CC3 approximation, Slater basis set

Table 2. Excitation energies of the calcium atom in ${\mathrm{c}\mathrm{m}}^{-1}$.

State	EXP [60,61]	CCSD(G)	CCSD(S)	CC3(G)	CC3(S)
$3{\mathrm{P}}^{\circ }$	15263.1	15098.7	15173.2	15063.5	15195.3
$3\mathrm{D}$	20356.6	–${}^{\mathrm{a}}$	20856.1	–${}^{\mathrm{a}}$	21299.6
$1\mathrm{D}$	21849.6	–${}^{\mathrm{a}}$	22878.6	–${}^{\mathrm{a}}$	22859.0
$1{\mathrm{P}}^{\circ }$	23652.3	24724.4	24845.8	24080.5	23879.6
$3\mathrm{S}$	31539.5	31518.3	31828.7	31157.3	31545.5
$1\mathrm{S}$	33317.3	33566.5	33890.9	32983.0	–${}^{\mathrm{a}}$

${}^{\mathrm{a}}$Iterations did not converge to the desired state. Explained in the text.

Table 3. Excitation energies of the strontium atom in ${\mathrm{c}\mathrm{m}}^{-1}$.

State	EXP [62]	CCSD(G)	CCSD(S)	CC3(G)	CC3(S)
$3{\mathrm{P}}^{\circ }$	14702.9	14575.6	14546.3	14570.8	14597.2
$3\mathrm{D}$	18253.8	18414.5	18155.0	18668.8	18393.7
$1\mathrm{D}$	20149.7	20814.1	20584.7	20650.3	20411.1
$1{\mathrm{P}}^{\circ }$	21698.5	22632.7	22701.9	21764.3	21797.5
$3\mathrm{S}$	29038.8	29137.0	29189.7	28885.3	28939.3
$1\mathrm{S}$	30591.8	31014.0	31063.1	30464.4	30508.6

Table 4. Excitation energies of the barium atom in ${\mathrm{c}\mathrm{m}}^{-1}$.

State	EXP [63]	CCSD(G)	CCSD(S)	CC3(G)	CC3(S)
$3\mathrm{D}$	9357.8	9270.9	8923.7	9581.6	9178.1
$1\mathrm{D}$	11395.4	12063.6	11653.5	11869.7	11391.4
$3{\mathrm{P}}^{\circ }$	13085.5	12970.5	12823.6	13069.8	12925.9
$1{\mathrm{P}}^{\circ }$	18060.3	19569.0	19527.3	18372.2	18284.6
$3\mathrm{S}$	26160.3	26136.6	26269.3	24275.7	26141.9
$1\mathrm{S}$	26757.3	27760.6	27971.9	25826.2	25213.0

Table 5. Lifetime τ in μs of the $5s5p3{\mathrm{P}}_1^{\circ }$ state of the Sr atom.

Reference	τ (μs)
Theory
This work XCCSD(G)	23.67
This work XCC3(G)	25.00
This work XCCSD(S)	23.24
This work XCC3(S)	24.60
Skomorowski et al. [57]	21.40
Santra et al. [64]	24.4
Porsev et al. [65]	19.0
Experiment
Zelevinsky et al. [66]	$21.5\pm 0.2$

Table 6. Lifetimes for the $5s6d3{\mathrm{D}}_2$ (s) state for barium atom. (T/E, L/V) denote Theoretical/Experimental energy and Length/Velocity representation.

Reference		Method	τ (s)
This work		XCC3(G)	20.0
Migdalek et al. [67]	$\{\begin{array}{l}\\ \\ \\ \\ \\ \\ \end{array}$	MCDF-I (T, L)	418.3
		MCDF-I (T, V)	3404.2
		MCDF-I (E, L)	582.6
		MCDF-I (E, V)	4153.0
		MCDF-II (T, L)	43.6
		MCDF-II (T, V)	50.6
		MCDF-II (E, L)	59.4
		MCDF-II (E, V)	60.9
Trefftz et al. [68]		MCHF	20.0

Table 7. Lifetime τ in ms of the $3d4s1{\mathrm{D}}_2$ state of the Ca atom.

Reference	τ (ms)
Theory
This work XCC3(S)	3.5
Bauschlicher et al. [69]	3.05
Bauschlicher et al. [69]	2.76
Experiment
Beverini et al. [70]	$2.3\pm 0.5$
Drozdowski et al. [71]	$1.5\pm 0.4$
Pasternack et al. [72]	$2.3\pm 0.5$

Table 8. Lifetime τ in ms of the $5s4d1{\mathrm{D}}_2$ state of the Sr atom.

Reference	τ (ms)
Theory
This work XCCSD(G)	0.43
This work XCC3(G)	0.52
This work XCCSD(S)	0.36
This work XCC3(S)	0.34
Skomorowski et al. [57]	0.23
Bauschlicher et al. [69]	0.49
Experiment
Courtillot et al. [73]	0.30
Husain and Roberts [74]	$0.41\pm 0.01$

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