ω j, l ¼ IP j þ ε j, l ;
ð41Þ
where IP j is the ionization potential of the core electron j (determined from the
energy differences of the (NÀ1)-electron ionic system and the N-electron neutral
system).
STEX is a single excitation theory in which particle and hole are independent.
Channel interaction can be included by diagonalizing the configuration interaction
matrix in the space of linear combinations of different single excitation channels.
This extension is known as the STEX-CIS (configuration interaction singles)
method. Double excitations can also be treated with a STEX Hamiltonian based
on the (NÀ2)-electron ionic system, but the SCF convergence problem is more
serious and the spin coupling schemes are complicated [66].
State-to-state transition dipoles are necessary in order to simulate the nonlinear
X-ray spectroscopy signals. Because the STEX orbitals are not orthogonal to the
MOs of the N-electron system, L€ owdin’s rule [67] may be used for calculating the
transition dipoles between two states with nonorthogonal single particle orbitals.
The transition dipoles for single excitations are
Ψ A
^
d
Ψ B
¼
X
N config:
m, n
a m b n
X
i, j
À1
ð Þ
iþ j d
mn
i j Minor S
mn
ð Þ i j ;
ð42Þ
where Ψ A,B are two excited states with nonorthogonal single particle orbitals, ^
d is
the transition dipole operator, and a m and b n are configuration interaction (CI)
coefficients for different single excitation configurations (m and n) of state A
and B, respectively.
d
mn
ij ¼
X
p, q
c
*
ip, m, A c jq, n, B
ð
ϕ
*
p
^
d ϕ q dτ;
ð43Þ
is the transition dipole matrix between the single excitation configurations m and n,
c ip,m,A and c j,q,n,B are MO coefficients for the configurations m and n of state A
and B, respectively, and
S
mn
i j ¼
X
k, l
c
*
ik, m, A c jl, n, B
ð
ϕ
*
i ϕ j dτ;
ð44Þ
is the overlap matrix between the MOs of the configurations m and n of state A
and B, respectively. ϕ i,j in (43) and (44) are basis functions and i, j, p, q, k, l are
indices for these basis functions. Minor (S
mn ) ij denotes the (i, j) minor of the matrix
S
mn .
We next present 1D and 2D SXRS signals calculated using STEX [29, 68]. Figure 9 shows the 2D-SXRS spectrum of N-methylacetamide (NMA) with the OOO
(O1s pump with two O1s probes) pulse sequence, together with 1D projections
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
297
ð41Þ
where IP j is the ionization potential of the core electron j (determined from the
energy differences of the (NÀ1)-electron ionic system and the N-electron neutral
system).
STEX is a single excitation theory in which particle and hole are independent.
Channel interaction can be included by diagonalizing the configuration interaction
matrix in the space of linear combinations of different single excitation channels.
This extension is known as the STEX-CIS (configuration interaction singles)
method. Double excitations can also be treated with a STEX Hamiltonian based
on the (NÀ2)-electron ionic system, but the SCF convergence problem is more
serious and the spin coupling schemes are complicated [66].
State-to-state transition dipoles are necessary in order to simulate the nonlinear
X-ray spectroscopy signals. Because the STEX orbitals are not orthogonal to the
MOs of the N-electron system, L€ owdin’s rule [67] may be used for calculating the
transition dipoles between two states with nonorthogonal single particle orbitals.
The transition dipoles for single excitations are
Ψ A
^
d
Ψ B
¼
X
N config:
m, n
a m b n
X
i, j
À1
ð Þ
iþ j d
mn
i j Minor S
mn
ð Þ i j ;
ð42Þ
where Ψ A,B are two excited states with nonorthogonal single particle orbitals, ^
d is
the transition dipole operator, and a m and b n are configuration interaction (CI)
coefficients for different single excitation configurations (m and n) of state A
and B, respectively.
d
mn
ij ¼
X
p, q
c
*
ip, m, A c jq, n, B
ð
ϕ
*
p
^
d ϕ q dτ;
ð43Þ
is the transition dipole matrix between the single excitation configurations m and n,
c ip,m,A and c j,q,n,B are MO coefficients for the configurations m and n of state A
and B, respectively, and
S
mn
i j ¼
X
k, l
c
*
ik, m, A c jl, n, B
ð
ϕ
*
i ϕ j dτ;
ð44Þ
is the overlap matrix between the MOs of the configurations m and n of state A
and B, respectively. ϕ i,j in (43) and (44) are basis functions and i, j, p, q, k, l are
indices for these basis functions. Minor (S
mn ) ij denotes the (i, j) minor of the matrix
S
mn .
We next present 1D and 2D SXRS signals calculated using STEX [29, 68]. Figure 9 shows the 2D-SXRS spectrum of N-methylacetamide (NMA) with the OOO
(O1s pump with two O1s probes) pulse sequence, together with 1D projections
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
297