order in the occupation number changes of the transition orbitals. The transition
state method is convenient and sufficiently accurate in many cases, but it is not
suitable for calculating many excited states because of the nonorthogonality and
self-consistent field (SCF) collapse of the excited states. An alternative is the
transition potential method (TP) [55, 56], in which the potential corresponding to
the transition hole state (a half electron on the hole orbital; see Fig. 7) is used to
produce a set of orthogonal excited states. The excitation energy is determined by
the differences between transition potential orbital energies. TP is widely used in
X-ray absorption spectroscopy simulation [57].
Similar core hole approximations have been proposed in solid state physics. The
half core hole approximation (HCH; see Fig. 7) is similar to the transition potential
method, and the full core hole approximation (FCH; see Fig. 7) is similar to the
direct exchange method (STEX) [58–60] in quantum chemistry (this is explained in
detail in the next subsection). In FCH, the impact of the excited electron on the core
hole is neglected. If it is included, we obtain the excited core hole approximation
(XCH) [61]. We have used XCH combined with TDDFT to simulate X-ray doublequantum-coherence spectroscopy [24].
3.1.2 Static Exchange Method (STEX)
In Hartree–Fock theory, occupied orbitals often provide an adequate description for
the ground state but virtual orbitals give a less satisfactory description of the excited
states. Hunt and Goddard proposed to use the Hartree–Fock virtual orbitals of an
(NÀ1)-electron system to represent the excited state orbitals of the corresponding
N-electron system. This is known as the improved virtual orbital or NÀ1 approximation [58]. In STEX, the occupied orbitals of an N-electron core-excited system
are also represented by the occupied orbitals of the (NÀ1)-electron ionic system
with the corresponding core hole. A core electron is removed and a restricted openshell Hartree–Fock (ROHF) calculation is carried out to obtain the occupied
orbitals of the ionic system. A major difficulty is that the electrons often collapse
to fill the core hole during the SCF calculation. This can be remedied by the
maximum overlap method (MOM) [62], which is explained in detail in the following sections. However, even with MOM, the SCF iteration may converge to a
wrong electronic state or even may not converge at all. To guide the SCF iteration
towards the designated ionic state, a careful choice of the other SCF convergence
parameters such as the damping and level-shifting factors [63–65] and many trialand-error calculations with different initial guesses are usually necessary. New
convergence schemes are required to improve the SCF calculations of such ionic
states.
Once the occupied orbitals of the ionic state are obtained, a single electron is
placed in a virtual orbital and the resulting open-shell singlet reads
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
295
state method is convenient and sufficiently accurate in many cases, but it is not
suitable for calculating many excited states because of the nonorthogonality and
self-consistent field (SCF) collapse of the excited states. An alternative is the
transition potential method (TP) [55, 56], in which the potential corresponding to
the transition hole state (a half electron on the hole orbital; see Fig. 7) is used to
produce a set of orthogonal excited states. The excitation energy is determined by
the differences between transition potential orbital energies. TP is widely used in
X-ray absorption spectroscopy simulation [57].
Similar core hole approximations have been proposed in solid state physics. The
half core hole approximation (HCH; see Fig. 7) is similar to the transition potential
method, and the full core hole approximation (FCH; see Fig. 7) is similar to the
direct exchange method (STEX) [58–60] in quantum chemistry (this is explained in
detail in the next subsection). In FCH, the impact of the excited electron on the core
hole is neglected. If it is included, we obtain the excited core hole approximation
(XCH) [61]. We have used XCH combined with TDDFT to simulate X-ray doublequantum-coherence spectroscopy [24].
3.1.2 Static Exchange Method (STEX)
In Hartree–Fock theory, occupied orbitals often provide an adequate description for
the ground state but virtual orbitals give a less satisfactory description of the excited
states. Hunt and Goddard proposed to use the Hartree–Fock virtual orbitals of an
(NÀ1)-electron system to represent the excited state orbitals of the corresponding
N-electron system. This is known as the improved virtual orbital or NÀ1 approximation [58]. In STEX, the occupied orbitals of an N-electron core-excited system
are also represented by the occupied orbitals of the (NÀ1)-electron ionic system
with the corresponding core hole. A core electron is removed and a restricted openshell Hartree–Fock (ROHF) calculation is carried out to obtain the occupied
orbitals of the ionic system. A major difficulty is that the electrons often collapse
to fill the core hole during the SCF calculation. This can be remedied by the
maximum overlap method (MOM) [62], which is explained in detail in the following sections. However, even with MOM, the SCF iteration may converge to a
wrong electronic state or even may not converge at all. To guide the SCF iteration
towards the designated ionic state, a careful choice of the other SCF convergence
parameters such as the damping and level-shifting factors [63–65] and many trialand-error calculations with different initial guesses are usually necessary. New
convergence schemes are required to improve the SCF calculations of such ionic
states.
Once the occupied orbitals of the ionic state are obtained, a single electron is
placed in a virtual orbital and the resulting open-shell singlet reads
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
295