138
A. Łachma´ nska et al.
L =
Φ 0
e
κ He
−κ
Ψ
AP1roG
el
+
i,a
λ
a
i
Φ
a ¯
a
i ¯
i
e
κ He
−κ
Ψ
AP1roG
el
− Ec
a
i
, (34)
where κ is again the generator of orbital rotations as defined in (36) and {λ
a
i } are the
Lagrange multipliers. The Lagrange multipliers are obtained from equations that are
analogous to the Λ-equations of coupled cluster theory, where we require the partial
derivative of the Lagrangian with respect to the geminal coefficients {c
a
i } to equal
zero,
∂L
∂c
a
i
| κ=0 = 0. The geminal coefficients are obtained by making L stationary
with respect to the Lagrange multipliers {λ
a
i },
∂L
∂λ
a
i
| κ=0 = 0 [20]. The orbital gradient
is the partial derivative of L with respect to the orbital rotation coefficients {κ pq }
evaluated for the current set of orbitals (κ = 0),
∂L
∂κ pq
κ=0
= g pq | κ=0 =
Φ 0 +
i,a
λ
a
i Φ
a ¯
a
i ¯
i
[(a
†
p a q − a
†
q a p ), H ]
Ψ
AP1roG
el
−
i,a
Φ 0
[(a
†
p a q − a
†
q a p ), H ]
Ψ
AP1roG
el
i,a
λ
a
i c
a
i . (35)
After the orbital gradient and (approximate) orbital Hessian A are evaluated, the
matrix representation of κ can be determined from
κ = −Ag
(36)
and the orbital basis can be transformed using the unitary transformation matrix e
−κ .
For reasons of computational efficiency, the orbital Hessian is typically approximated
by its diagonal, A pq, pq =
∂ g pq
∂κ pq
κ=0
.
Although AP1roG captures a significant amount of the strong electron correlation
energy and represents a very promising reference wave function in actinide chemistry,
it misses a large fraction of the dynamic (weak) correlation energy. Dynamic electron
correlation effects on top of the geminal wave function can be included in the wave
function ansatz a posteriori using, for instance, perturbation theory [70, 95, 120, 127,
141], extended random phase approximation [27, 47, 109, 114], density functional
theory (DFT) [44, 45, 47], and coupled cluster theory [136] or its linearized version
[13, 47, 88]. Numerical studies indicate that the perturbation theory corrections
with an AP1roG reference function do not provide reliable electronic structures and
properties for actinide-containing compounds. To reliably model actinide chemistry,
we can use various coupled cluster corrections on top of the AP1roG wave function.
One possible way to extend AP1roG is to apply an AP1roG-tailored CC formalism. In AP1roG-tailored CC theory, the electron-pair amplitudes of the CC singles,
doubles, triples, etc. equations are substituted by the AP1roG geminal coefficients
and not optimized, that is kept frozen, during the optimization procedure. Note that
the tailored CC amplitude equations are similar to the conventional CC working equations, except that some selected amplitudes (the tailored amplitudes) are not varied.
Thus, a tailored CC calculation represents only a minor modification in any CC code.
A. Łachma´ nska et al.
L =
Φ 0
e
κ He
−κ
Ψ
AP1roG
el
+
i,a
λ
a
i
Φ
a ¯
a
i ¯
i
e
κ He
−κ
Ψ
AP1roG
el
− Ec
a
i
, (34)
where κ is again the generator of orbital rotations as defined in (36) and {λ
a
i } are the
Lagrange multipliers. The Lagrange multipliers are obtained from equations that are
analogous to the Λ-equations of coupled cluster theory, where we require the partial
derivative of the Lagrangian with respect to the geminal coefficients {c
a
i } to equal
zero,
∂L
∂c
a
i
| κ=0 = 0. The geminal coefficients are obtained by making L stationary
with respect to the Lagrange multipliers {λ
a
i },
∂L
∂λ
a
i
| κ=0 = 0 [20]. The orbital gradient
is the partial derivative of L with respect to the orbital rotation coefficients {κ pq }
evaluated for the current set of orbitals (κ = 0),
∂L
∂κ pq
κ=0
= g pq | κ=0 =
Φ 0 +
i,a
λ
a
i Φ
a ¯
a
i ¯
i
[(a
†
p a q − a
†
q a p ), H ]
Ψ
AP1roG
el
−
i,a
Φ 0
[(a
†
p a q − a
†
q a p ), H ]
Ψ
AP1roG
el
i,a
λ
a
i c
a
i . (35)
After the orbital gradient and (approximate) orbital Hessian A are evaluated, the
matrix representation of κ can be determined from
κ = −Ag
(36)
and the orbital basis can be transformed using the unitary transformation matrix e
−κ .
For reasons of computational efficiency, the orbital Hessian is typically approximated
by its diagonal, A pq, pq =
∂ g pq
∂κ pq
κ=0
.
Although AP1roG captures a significant amount of the strong electron correlation
energy and represents a very promising reference wave function in actinide chemistry,
it misses a large fraction of the dynamic (weak) correlation energy. Dynamic electron
correlation effects on top of the geminal wave function can be included in the wave
function ansatz a posteriori using, for instance, perturbation theory [70, 95, 120, 127,
141], extended random phase approximation [27, 47, 109, 114], density functional
theory (DFT) [44, 45, 47], and coupled cluster theory [136] or its linearized version
[13, 47, 88]. Numerical studies indicate that the perturbation theory corrections
with an AP1roG reference function do not provide reliable electronic structures and
properties for actinide-containing compounds. To reliably model actinide chemistry,
we can use various coupled cluster corrections on top of the AP1roG wave function.
One possible way to extend AP1roG is to apply an AP1roG-tailored CC formalism. In AP1roG-tailored CC theory, the electron-pair amplitudes of the CC singles,
doubles, triples, etc. equations are substituted by the AP1roG geminal coefficients
and not optimized, that is kept frozen, during the optimization procedure. Note that
the tailored CC amplitude equations are similar to the conventional CC working equations, except that some selected amplitudes (the tailored amplitudes) are not varied.
Thus, a tailored CC calculation represents only a minor modification in any CC code.