Contents

 

1. Determining an absolute energy scale for the first excited state.

2. Improved relative energies between excited states

3. Open-shell ground state

4. Functional dependence

5. Broadening

 

1. Absolute energy scale for the first excited state.

 

This requires three calculations for each spectrum contribution,

       the standard TP calculation which gives the overall spectrum with relative energies given through the orbital energies and oscillator strengths from the TP orbitals. This corresponds to a balanced contribution of initial and final state effects. (See also discussion in Cavalleri et al., Phys. Chem. Chem. Phys. 7, 2854 (2005)).

       the ground state calculation without the core-hole.

       the first core-excited state calculation using keyword EXCITED as for the TP calculation but with occupation 0.0 in the core level of appropriate spin and one extra occupied orbital (the LUMO from the ground state calculation) added of the same spin. This accounts for a more accurate description (with respect to the TP approach) of electronic relaxation for the specific excitation, both the response to the core hole and the presence of the excited electron.

The absolute energy position of the first core-excited state is obtained as the difference in total energy (with numerically integrated exchange-correlation energy) between the ground state and the first core-excited state.

 

In addition a correction for relativistic effects can be added to improve the position (see [Takahashi and Pettersson, J. Chem. Phys. 121, 10339 (2004)]). This is a purely atomic effect and can be added separately. The remaining differences from experiment are then due to the functional used, incompleteness in the basis set as well as possible deficiencies in the model used to represent the experimental situation; having a computed reliable absolute energy scale helps to identify and isolate the latter possibility. The functional dependence will be discussed under point 4 below.

 

Specific example: First core-excited state (1s 4a1) of gas phase water in C2v symmetry.

 

Ground state:

 

FSYM SCFOCC

ALFA 3 1 1 0

BETA 3 1 1 0

 

Transition potential:

 

FSYM SCFOCC EXCITED

ALFA 3 1 1 0

BETA 3 1 1 0

SYM 1

ALFA 0 1 1 0.5

BETA 0 0

END

 

First core-excited:

 

FSYM SCFOCC EXCITED

ALFA 4 1 1 0

BETA 3 1 1 0

SYM 1

ALFA 0 1 1 0.0

BETA 0 0

END

 

2. Improved relative energies between excited states

 

Here we generate a sequence of mutually orthogonal core-excited states by extending the above procedure. We view the Kohn-Sham state as representable by a single determinant which means that if we remove the LUMO, as defined using the procedure above, from the variational space and instead occupy the LUMO+1 orbital we will obtain a variational representation of the next core-excited level; the difference in one orbital (the LUMO which is orthogonal to every other orbital by this construction) guarantees the orthogonality of the two states. This is actually in agreement with the Hohenberg-Kohn theorem which uses the variational principle for the proof. It is essential here to do a restart from the previous calculation defining the orbitals for the first core-excited state.

 

This procedure takes into account possible differences in the response of the molecular ion core to the actual presence of the excited electron in specific excited orbitals; in the basic TP formalism the molecular ion core is treated as a frozen density. The steps below can be useful when specific low-lying excited states have different character and thus affect the molecular ion core differently. Examples where this has been used can be found in, e.g., [Kolczewski et al., J. Chem. Phys. 115, 6426 (2001); Cavalleri et al., Phys. Chem. Chem. Phys. 7, 2854 (2005)].

 

Specific example: 5a1, 6a1 and 2b2 states of gas phase water in C2v symmetry.

 

1s5a1 state: (Use EXCITED to assign zero occupation to 1s and 4a1 and SUPSYM to ensure that 4a1 cannot mix with the other orbitals. This calculation must start from the 4a1 restart file using the NEWOCC keyword.)

 

RUNTYPE NEWOCC

FSYM SCFOCC EXCITED

ALFA 5 1 1 0

BETA 3 1 1 0

SYM 1

ALFA 0 2 1 0.0 4 0.0

BETA 0 0

END

SUPSYM

1

4

END

 

1s6a1 state: (Use EXCITED to assign zero occupation to 1s, 4a1 and 5a1 and SUPSYM to ensure that they cannot mix with the other orbitals. This calculation must start from the 5a1 restart file using the NEWOCC keyword.)

 

RUNTYPE NEWOCC

FSYM SCFOCC EXCITED

ALFA 6 1 1 0

BETA 3 1 1 0

SYM 1

ALFA 0 3 1 0.0 4 0.0 5 0.0

BETA 0 0

END

SUPSYM

1

4

1

5

END

 

1s2b2 state: (Use EXCITED to assign zero occupation to 1s. Since 2b2 is already in a separate symmetry from 4a1 and 5a1 the b2 states can be treated as a separate sequence and a normal start made since this is the lowest state in its symmetry. SUPSYM is not necessary only if higher states of the same symmetry are required.)

 

RUNTYPE START

FSYM SCFOCC EXCITED

ALFA 3 2 1 0

BETA 3 1 1 0

SYM 1

ALFA 0 1 1 0.0

BETA 0 0

END

 

3. Open-shell ground state

 

Assuming that the orbital occupations in the two spin symmetries are different then excitations of ALFA and BETA spin must be considered separately including the corrections, i.e. three calculations (TP, GS and 1st core-excited) for each of the two spins if an absolute energy scale is required. This is done most easily by either keeping an ALFA hole and switching occupations or moving the hole to BETA symmetry.

 

4. Functional dependence

 

This has been investigated and discussed in [Takahashi and Pettersson, J. Chem. Phys. 121, 10339 (2004)] where we found that most of the dependence on the functional is associated with the 1s level where the electron density is the highest and that it mainly leads to a shift in the 1s ionization potential (IP). Core IPs and the first core-excited state (energy difference between GS and first core-excited includes removal of 1s) thus exhibit a dependence in their energy positions on the functional used, while excitation energy differences (term values) are much less affected; here the influence of the 1s all but cancels out.

 

Though not strictly transferable this opens the possibility for an empirical correction to a very accurate absolute energy scale if a (very) similar reference system can be found for which the core IP is experimentally known. This is the case for, e.g., water where H-bonding is not expected to modify strongly the behavior of the 1s. Here we can thus calculate the gas phase core IP and introduce an empirical correction to the calculated core-excitation energy (defined as in 1 and 2 above) based on the difference between experiment and theory. Note that this then already includes the effects of relativity and in addition corrects for basis set deficiencies in the description of the core orbital.

 

This procedure has been used in [Leetmaa et al., J. Chem. Phys. 125, 244510 (2006)].

 

Note, however, that according to the study in [Takahashi and Pettersson, J. Chem. Phys. 121, 10339 (2004)] we cannot define a general correction for each element; there are also effects of the chemical environment.

 

5. Broadening

 

This is in a way cosmetics and the intention with the broadening is very much to be able to communicate in a common language with experimentalists. The program generates oscillator strengths (cross sections) at specific energies and the aim is to transfer this into a continuous spectrum. For gas phase molecules an alternative procedure is given by Stieltjes imaging but since this is dependent on the IP it is not applicable to solid state problems where the IP is ill-defined or not defined at all.

 

There are no uniquely valid procedures when it comes to the broadening but we recommend the following:

 

a) Just plot initially the oscillator strengths with a constant broadening, i.e. 0.5 eV for all states as a control (see, e.g., [Cavalleri et al., Phys. Chem. Chem. Phys. 7, 2854 (2005); Odelius et al., Phys. Rev. B. 73, 024205 (2006)]). The broadening should not change significantly the appearance of this spectrum; in particular not move intensity to lower energy (the focus is on the near-edge features).

 

b) Use utility xrayspec to broaden the spectrum applying a fixed smaller broadening before the edge (or a suitable onset) and then increase linearly the broadening to a constant larger fwhm. The interval over which the broadening is increased should be large enough such that intensity does not spill over from the end-point to before the starting point. Since a Gaussian broadening is used this means an interval of the order 3x fwhm. The final broadening should be large enough that spurious basis set dependent states in the continuum are not mistaken for real resonances; these appear as a result of the sum rules (the sum of oscillator strengths is related to the probability of absorption) and the limited sampling of the continuum using the local augmentation basis set. In this way an isolated strong oscillator strength is smeared out while a region with several transitions correctly still indicates a resonance in the continuum. This can, however, be further investigated according to the procedure under c) next.

 

c) When in doubt about the validity of the broadening and whether a continuum resonance is real or an artifact, the augmentation basis can be further extended stepwise to converge the representation further away from the edge than afforded by the standard augmentation basis set. This requires only the restart file from the XRAY (TP) calculation.

 

As an example we assume that O1 is the central oxygen in a water cluster, O2 the nearest neighbors, O3 further away etc. The initial TP calculation for O1 with the order given through:

 

O1

O2

O3

O4

O5

 

would then be:

 

RUNTYPE START

...

XRAY XAS

END

...

X-FIRST

X-DUMMY

X-DUMMY

X-DUMMY

X-DUMMY

 

The next would add O2 but performed as a property calculation only, based on the TP restart file (defines the potential):

 

RUNTYPE CPRO

...

XRAY XAS

END

...

X-FIRST

X-FIRST

X-DUMMY

X-DUMMY

X-DUMMY

 

This procedure can then be continued (until the program runs out of memory) and will successively improve the description going higher and higher up in energy; this can be used to investigate whether an indicated resonance in the continuum is real or not and also to give some idea of how far from the edge that the computed spectrum is stable with respect to the basis set.

 

We do hope that these instructions will help all StoBe users that are performing XAS calculations. When in doubt or in need of confirmation of the soundness of a computational model you are welcome to contact the StoBe authors.