Investigations of Rydberg-Valence mixing in the MC-srDFT
computational chemistry model
Diana Kjølby and Hans Jørgen Aagaard Jensen
Department of Physics, Chemistry and Pharmacy,
University of Southern Denmark, DK-5230 Odense M, Denmark
(Dated: 28-Aug-2016)
In Hubert, Hedegård, & Jensen (2016) it was found that the multiconfiguration
short-range density functional theory (MC-srDFT) method can perform well for valence electronic excitations for a wide range of organic molecules, including the nucleobases. In this paper we show that the MC-srDFT method can also properly describe
Rydberg states and, related, avoid spurious Rydberg-valence mixing when the used
basis sets are augmented with Rydberg basis functions.
TO DO:
1. add missing column(s) to tables
2a. change bibliography into a bibtex type
2b. change literature references to \cite
3. create labels for all tables and figures and use these in all references
4. use latex references to sections and subsections so we get the right numbers
5. add appropriate references to experimental data (compare reference paper)
2
TABLE OF CONTENTS
CONTENTS
Table of contents
2
Glossary
3
I. Introduction
A. Rydberg-Valence Mixing
II. Theory
4
4
5
A. Rydberg transitions
5
B. The MC-srDFT method
6
III. Computational Details
7
A. Molecules
7
B. Basis sets
8
IV. Results and discussion
A. MC-srDFT methods and added Rydberg basis functions in the basis set
8
8
B. Evaluation of choice of range for neff
10
C. Ionization potential
12
D. Investigation of maximum ` for added Rydberg basis functions
14
E. The influence of MC-srDFT
15
V. Conclusions
VI. References
19
20
3
GLOSSARY
MC-srDFT: Multi-configuration self-consistent field short-range Density Functional Theory
KS-DFT: Kohn-Sham Density Functional Theory
MCSCF: Multi-configurational Self-consistent Field
TZVP: Triple-Zeta Valence Polarized
PBE:
CAS-srPBEGWS:
HF-srPBEGWS:
4
I.
INTRODUCTION
This research paper will investigate the Rydberg-Valence mixing of the MC-srDFT computational chemistry method. Smaller molecules all have both Rydberg, valence and mixing
type states. The Rydberg-Valence mixing can cause significant errors in the computed
UV/vis spectra. The MC-srDFT method has shown potential for electron excitations. Thus
it is of great interest to investigate MC-srDFT performance in this matter. The investigation
is performed by calculating excited states for some molecules which have shown RydbergValence mixing. The MC-srDFT method is applied with various basis set. The goal for
this investigation is to make MC-srDFT more useful to the future user. This is done by informing the user about it’s ability to make a satisfactory result for molecules with spurious
Rydberg-valence mixing.
A.
Rydberg-Valence Mixing
The definition of Rydberg states is when an electron is excited from the valence shell into
an orbital of higher principal quantum number (Atkins & Friedman, 2005). Every molecule
has Rydberg states. The state is called Rydberg if it is of higher quantum number than the
groundstate, CITATION Ger66 \l 1030 (Molecular spectra and molecular structure, 1966).
When using Ab-Intio CI calculations, problems occur with the description of the Rydberg
states and in specific their interactions with neighboring valence states. The mixing can
occur when the valence and Rydberg states are close in energy level. When working with
Rydberg states the basis set is highly important. It is necessary to use a diffuse basis set
since Rydberg states by definition is diffuse (Buenker, Hirsch, & Li). The published paper
(Hubert, Hedegård, & Jensen, 2016) has investigated MC-srDFT for electronic excitations
in organic molecules and concluded that the issue of Rydberg-valence mixing needs more
research. Since the description of Rydberg-valence mixing was not always correctly described
by the srDFT functionals of srPBE type. Especially high excitation energies are expected
to be more sensitive to the method and basis set applied in the calculations. This is caused
by Rydberg-Valence mixing.
5
II.
THEORY
This section will give a short overview of Rydberg transitions. Furthermore some insight
in the theory behind the MC-srDFT method including the choice of the parameter µ.
A.
Rydberg transitions
The Rydberg series converges to the ionization potential, which is defined as the energy
required to remove the electron from the ionic core. According to (Rydberg transitions,
2000) a Rydberg transition is a sharp transition the frequency of which can be described by
a Hydrogen-like formula:
hν = i −
R
R
= i −
2
(n − δ` )
(neff )2
(1)
where i is the ionization potential to which the series converges, R is the Rydberg constant,
n is the principal quantum number, and δ` is the quantum defect. The angular symmetry `
of the Rydberg series can experimentally be deduced from the fitted value of δ` : δ0 ≈ 1 for
an s series, δ1 ≈ 0.6 for a p series, and δ2 ≈ 0.1 for a d series.
A problem is that a Rydberg transition is not always sharp but broad and does not fit
the description in Equation (1). Which is why the valence-Rydberg mixing occurs because a
broad transition might as well be a valence transition. The valence will also fit the equation,
though it does not belong to a series with increasing quantum numbers n like Rydberg
states. An electron in a polyatomic molecule will fit, if it is diffuse, because the electron
may have a large radius and thus described as a Rydberg state. This will not be the case
for neutral polyatomic molecules.
Regarding oscillator strengths it is observed that the lowest Rydberg transitions never
exceed 0.08 compared to valence shell transitions which is easily seen to be 0.1-0.3. In
addition, the mixing will give too high excitation energy. This can be improved by adding
the Rydberg orbital descriptions. Then the result will improve the match to the experimental
data.
6
B.
The MC-srDFT method
The MC-srDFT method is described by (Hubert, Hedegård, & Jensen, 2016) as a method
which is a multi-configurational extention of the KS-DFT method. The method relies on
separation of the range for two-electron repulsion. The coulomb interaction is separated into
a long-range and a short-range component:
1
sr,µ
lr,µ
(r12 ) ,
(r12 ) + wee
= wee
r12
(2)
sr,µ
lr,µ
(r12 ) is the short-range
(r12 ) is the long-range two-electron repulsion and wee
where wee
two-electron repulsion. The parameter µ controls the range separation. This splitting describes the long-range interaction by the error function in
lr,µ
wee
(r12 ) =
erf(µ r12 )
r12
(3)
Then to describe the ground energy it is known from the variation principle that it is always
less or equal the expectation value. Together with the long-range and short-range separation
Equation (2) and Equation (3) it will produce the energy for the ground state:
E
nD o
sr,µ
µ
lr,µ µ
E = min
Ψ T̂ + V̂ne + Ŵee Ψ + EHxc [ρΨµ ] .
µ
µ
Ψ
(4)
The energy is expressed as a functional of the charge density. This is a generalization of
the KS-DFT. Notice that the nuclear-electron attraction part in (4) can be written as an
operator since it is one-electron dependent. In difference to the -dependent short-ranged
Hartree-exchange-correlation functional which can’t be written as a linear operator. This
results in equation (5).
(5)
Equation (5) describes the MC-srDFT energy as a generalization of the MCSCF wave function. In the equation is the standard non-relativistic kinetic energy operator, is the standard
nuclear-electron attraction potential, and is the long-range electron-electron repulsion operator. The general idea in the MC-srDFT method is that the switching parameter µ can
be used to provide a better computational model than pure KS-DFT and pure MCSCF (the
µ = 0 limit DFT and the µ → ∞ limit, respectively)
The aim is also to have lower computational cost while achieving a satisfactory result.
An optimal range for µ was found by (Fromager, Toulouse, & Jensen, 2007) given by the
7
restriction
0.33 bohr−1 ≤ µ ≤ 0.5 bohr−1 .
(6)
The MC-srDFT method used for the calculations in this paper is an extension to the timedependent regime. For further details see (Fromager, Toulouse, & Jensen, 2007).
III.
COMPUTATIONAL DETAILS
In this section a short overview of the molecules and basis sets used in the calculations
will be provided.
For the computations with the hybrid CAS-srPBEGWS and HF-srBPEGWS methods we
used µ = 0.4 as in (Hubert, Hedegård, & Jensen, 2016), the reference results for this work.
REMEMBER TO INSERT DESCRIPTION of srPBEGWS with appropriate references.
A.
Molecules
The molecules chosen for this investigation are extracted from (Hubert, Hedegård, &
Jensen, 2016) because these molecules had low-lying extitation found to be dependent on
the basis set, presumably due to Rydberg-Valence mixing. The molecules chosen are the six
organic molecules: formaldehyde, acetone, acetamide, cytosine, thymine and uracil. These
are illustrated in Figure 1.
FIG. 1. The six organic molecules chosen for the investigation.
8
B.
Basis sets
The reference basis set used for the molecules is the Ahlrichs triple-zeta valence polarized
(TZVP) basis set//REF// in order to relate the results to the paper (Hubert, Hedegård, &
Jensen, 2016). For a description of the Rydberg states center-of-mass basis functions is used
as proposed by Kaufmann et al. (Karl Kaufmann, 1989). Three parameters are required to
specify these Rydberg basis function: maximum angular quantum number `, minimum value
for neff , and finally maximum value for neff , cf. Equation (1). Following (Karl Kaufmann,
1989) we employ half-integers starting and ending quantum numbers neff for the Rydberg
basis functions. This is expected to give satisfactory accuracy for all relevant values of the
quantum defect parameter δ` .
IV.
RESULTS AND DISCUSSION
In this section the results from the investigation will be discussed. The results will be
compared to the experimental results from (Kenneth B. Wiberg, 2002). First the Rydberg
basis functions will be added to the basis set and the effect of this will be evaluated. The
excited states from the MC-srDFT with the basis set TVZP will be shown next to the excited
states with the added Rydberg functions in subsection 4.1. In subsection 4.2 the influence
of the range for neff will be investigated. The minimum and maximum values for neff will
be evaluated in order to give a satisfactory match to the experimental data, while having a
fair computational cost. In subsection 4.3 the ionization potentials will be included in the
calculation. In subsection 4.4 the influence of maximum quantum number of the Rydberg
basis functions will be evaluated by testing the result with the maximum quantum number
of s, sp, spd, sdpf and spdf g in the Rydberg basis functions. Then to validate that the
MC-srDFT method is the reason of these satisfactory results, the calculations will be tested
with other methods as well, in subsection 4.5.
A.
MC-srDFT methods and added Rydberg basis functions in the basis set
The MC-srDFT method with the TZVP basis set is used to calculate the first 15 excited
states for the six organic molecules; formaldehyde, acetone, acetamide, cytosine, thymine
and uracil. To illustrate an example, the molecule formaldehyde has been selected and
9
TABLE I. The first 15 excited states for formaldehyde using CAS-srPBEGWS without and with
added Rydberg basis functions. The excited states vertical and the methods and symmetries horizontally. Excitation energies in eV and oscillator strengths in parentheses. Experimental data in
last row.
CAS-srPBEGWS (µ = 0.4)
Method
Basis Set
TZVP
TZVP + RYD-spd=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
8.05 (0.000)
8.56 (0.082)
8.96 (0.001)
3.81
8.07 (0.000)
7.20 (0.018)
8.95 (0.001)
3.80
Exc. 2
9.54 (0.067)
9.50 (0.056)
12.08 (0.023)
9.93
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
Exc. 3
10.57 (0.320)
12.54 (0.034)
13.17 (0.081)
12.08
9.17 (0.018)
8.98 (0.014)
9.83 (0.000)
9.25
Exc. 4
13.87 (0.008)
14.40 (0.002)
15.22 (0.084)
12.13
9.44 (0.006)
9.13 (0.001)
10.14 (0.000)
9.50
Exc. 5
14.23 (0.012)
14.53 (0.344)
16.18 (0.347)
13.68
9.56 (0.110)
9.22 (0.001)
10.42 (0.000)
9.86
Exc. 6
14.70 (0.416)
15.31 (0.031)
17.33 (0.077)
17.20
9.90 (0.018)
9.41 (0.008)
10.84 (0.044)
9.91
Exc. 7
15.65 (0.001)
15.72 (0.105)
17.97 (0.049)
18.06
9.95 (0.008)
9.73 (0.007)
11.00 (0.000)
9.97
Exc. 8
15.75 (0.003)
17.44 (0.000)
20.39 (0.000)
19.77
10.17 (0.008)
9.81 (0.000)
11.65 (0.026)
10.16
Exc. 9
17.08 (0.000)
18.28 (0.013)
21.09 (0.010)
20.66
10.20 (0.003)
9.85 (0.000)
12.13 (0.000)
10.21
Exc. 10 17.69 (0.057)
18.68 (0.010)
22.05 (0.017)
22.00
10.46 (0.014)
9.93 (0.003)
12.59 (0.004)
10.46
Exc. 11 19.03 (0.066)
18.89 (0.000)
22.39 (0.001)
23.56
10.49 (0.000)
10.08 (0.004)
12.78 (0.021)
10.50
Exc. 12 19.31 (0.212)
19.47 (0.200)
22.70 (0.000)
25.89
11.03 (0.030)
10.12 (0.000)
12.90 (0.004)
11.07
Exc. 13 19.84 (0.189)
20.31 (0.016)
24.57 (0.031)
26.36
11.13 (0.000)
10.15 (0.000)
13.07 (0.004)
11.14
Exc. 14 20.40 (0.055)
21.85 (0.009)
25.17 (0.000)
26.40
11.95 (0.013)
10.20 (0.002)
13.34 (0.013)
11.73
Exc. 15 20.70 (0.000)
22.95 (0.000)
25.44 (0.002)
26.65
12.35 (0.021)
10.29 (0.003)
13.36 (0.000)
12.09
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
8.14, 9.58
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
Exp.
8.14, 9.58
9.26, 9.63
9.26, 9.63
shown in Table 1.
The calculations with the basis set ”TZVP” (left side of Table 1) are compared to the
calculation with the added Rydberg basis functions “TZVP + Ryd-spd=2.5-5” (right side of
Table 1). Later the specifications of the Rydberg function are evaluated but for now notice
the effect of the added Rydberg functions in the MC-srDFT method. The idea is to make a
better description of the Rydberg states such the valence-Rydberg mixing will be minimized
as much as possible. Similar calculations have been performed for all six molecules presented
in Appendices A and B.
The first thing to notice is that the inclusion of Rydberg basis functions results in a lot of
Rydberg states within the lowest 15 excited states states compared to the values calculated
without the Rydberg basis functions. The values highlighted blue are those which are close
10
to the expected values presented in the last row of Table 1. Recall that that the valence
states will have oscillator strengths of 0.1-0.3 while the oscillator strengths of the Rydberg
states never exceed 0.080. It is clearly seen that with the Rydberg basis functions there
is a satisfactory estimate of all the experimental data found for formaldehyde. This makes
sense since the basis set TZVP did not cover the Rydberg states and with the Rydberg basis
functions the Rydberg states are better described and there occur less mixing. Hence a better
match for the experimental data. Similar tendencies and results are shown in Appendix A
for acetone and likewise in Appendix B for acetamide, cytosine, thymine and uracil. The
MC-srDFT method together with a proper basis set provides satisfactory results.
B.
Evaluation of choice of range for neff
To investigate how short a range for neff one can choose and still get good results, see
the “TZVP + Ryd-spd=3-4” (left side in Table 2) compared with “TZVP + Ryd-spd=2.5-4”
(right side in Table 2). We recall that all half-integer values from the lower limit to the
upper limit are included. First the upper limit is fixed at neff = 4 and the lower limits
neff = 2.5 and neff = 3 are compared. In Table 3 the lower limit is fixed and upper limits
neff = 4 and neff = 5 are compared.
In the example of formaldehyde it becomes clear that the lowering the lower limit for neff
from 3 to 2.5 has significant influence on the calculated excitation energies. The excitation
energies with a difference from 0.03 to 0.06 eV are highlighted green. The excitation energies
with a difference between 0.07-0.10 eV are highlighted blue. Finally the excited states that
change more than 0.10 eV is highlighted yellow. When the energies change by the shift from
one basis set to another then it is possibly due to the Rydberg-valence mixing. The energies
differs a lot from each other in the higher excitation states. There is seen a significant shift
in the calculated excitation energies by changing the lower limit from 3 to 2.5. Even the
lowest excitation energy change with more than 0.10 eV. Hence the lower limit for neff should
be set at 2.5.
Then to investigate a favored value for the upper limit for neff the method “TZVP +
Ryd-SPD=2.5-4” (left side of Table 3) is compared to “TZVP + Ryd-SPD=2.5-5” (right
side of Table 3). These have different upper limits for neff . The energy changes are again
highlighted with the colors: Green for small change 0.03-0.06 eV, Blue for change 0.07-0.10
11
TABLE II. The first 15 excited states for formaldehyde using CAS-srPBEGWS with added Rydberg
basis functions with different lower limits for neff . Excitation energies in eV and oscillator strengths
in parentheses.
TZVP + Ryd-spd=3-4
TZVP + Ryd-spd=2.5-4
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
8.06 (0.000)
7.38 (0.022)
8.95 (0.001)
3.81
8.07 (0.000)
7.21 (0.018)
8.95 (0.001)
3.80
Exc. 2
8.24 (0.033)
8.10 (0.032)
9.23 (0.000)
8.40
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
Exc. 3
9.18 (0.011)
9.04 (0.015)
9.89 (0.000)
9.26
9.17 (0.018)
8.99 (0.015)
9.84 (0.000)
9.25
Exc. 4
9.50 (0.000)
9.18 (0.001)
10.48 (0.000)
9.55
9.44 (0.006)
9.13 (0.001)
10.48 (0.000)
9.50
Exc. 5
9.59 (0.110)
9.32 (0.003)
11.02 (0.033)
9.89
9.56 (0.110)
9.23 (0.001)
10.85 (0.043)
9.87
Exc. 6
9.94 (0.026)
9.46 (0.007)
11.79 (0.025)
9.93
9.93 (0.024)
9.41 (0.008)
11.65 (0.016)
9.91
Exc. 7 10.00 (0.010)
9.76 (0.008)
12.65 (0.003)
10.03
9.98 (0.007)
9.73 (0.008)
11.67 (0.010)
10.02
Exc. 8 10.55 (0.023)
9.85 (0.001)
12.84 (0.010)
10.53
10.55 (0.033)
9.81 (0.000)
12.59 (0.004)
10.54
Exc. 9 12.02 (0.198)
9.91 (0.000)
12.98 (0.006)
11.87
10.65 (0.003)
9.86 (0.000)
12.78 (0.020)
10.69
Exc. 10 12.08 (0.010)
9.98 (0.006)
13.13 (0.006)
12.09
11.92 (0.034)
9.97 (0.004)
12.90 (0.004)
11.73
Exc. 11 12.64 (0.012)
10.23 (0.013)
13.39 (0.001)
12.87
12.03 (0.009)
10.15 (0.001)
13.07 (0.004)
11.94
Exc. 12 12.92 (0.002)
10.37 (0.002)
13.47 (0.007)
13.16
12.46 (0.000)
10.32 (0.010)
13.34 (0.013)
12.10
Exc. 13 13.13 (0.118)
10.82 (0.021)
13.51 (0.005)
13.46
12.91 (0.030)
10.39 (0.000)
13.37 (0.000)
12.87
Exc. 14 13.22 (0.000)
11.24 (0.029)
13.57 (0.003)
13.53
12.99 (0.064)
10.65 (0.011)
13.47 (0.007)
13.09
Exc. 15 13.57 (0.000)
12.87 (0.005)
13.64 (0.002)
13.66
13.16 (0.002)
11.14 (0.018)
13.53 (0.002)
13.52
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
8.14, 9.58
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
Exp.
8.14, 9.58
9.26, 9.63
9.26, 9.63
eV, Yellow change more than 0.10 eV.
Notice that the first five excitations in each symmetry does not change significantly except
from two in symmetry B2 . In this matter it is up to the user to decide if it is necessary
to increase the upper limit from 4 to 5. If the result ought to match the experimental
data then the enlarged basis set is irrelevant in our case, since the first values does not
change significantly. Thereby it would just be an extra cost to enlarge the basis set. The
“TZVP+Ryd-SPD=2.5-4” demanded 42 minutes 16 seconds in dalton and the “TZVP + RydSPD=2.5-5” demanded 56 minutes 42 seconds. The time in the example of formaldehyde
does not make a big difference but it could be worth considering for a larger DNA molecule.
Where the calculation time for just one cytosine is 11 h 32 min 28 s for upper limit of 4
and then 13 h 54 min 9 s for upper limit of 5. It is important to notice that the excitation
energies for thymine and uracil are unchanged from an upper limit of 4 to 5, see Appendix
B. This means that the only difference in the methods are the cost in time. In this case
12
TABLE III. The lowest 15 excited states in each symmetry for formaldehyde using the CASsrPBEGWS method with added Rydberg basis functions with two different upper limits for neff : 4
and 5. Excitation energies in eV and oscillator strengths in brackets.
TZVP + Ryd-spd=2.5-4
TZVP + Ryd-spd=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
8.07 (0.000)
7.21 (0.018)
8.95 (0.001)
3.80
8.07 (0.000)
7.20 (0.018)
8.95 (0.001)
3.80
Exc. 2
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
Exc. 3
9.17 (0.018)
8.99 (0.015)
9.84 (0.000)
9.25
9.17 (0.018)
8.98 (0.014)
9.83 (0.000)
9.25
Exc. 4
9.44 (0.006)
9.13 (0.001)
10.48 (0.000)
9.50
9.44 (0.006)
9.13 (0.001)
10.14 (0.000)
9.50
Exc. 5
9.56 (0.110)
9.23 (0.001)
10.85 (0.043)
9.87
9.56 (0.110)
9.22 (0.001)
10.42 (0.000)
9.86
Exc. 6
9.93 (0.024)
9.41 (0.008)
11.65 (0.016)
9.91
9.90 (0.018)
9.41 (0.008)
10.84 (0.044)
9.91
Exc. 7
9.98 (0.007)
9.73 (0.008)
11.67 (0.010)
10.02
9.95 (0.008)
9.73 (0.007)
11.00 (0.000)
9.97
Exc. 8 10.55 (0.033)
9.81 (0.000)
12.59 (0.004)
10.54
10.17 (0.008)
9.81 (0.000)
11.65 (0.026)
10.16
Exc. 9 10.65 (0.003)
9.86 (0.000)
12.78 (0.020)
10.69
10.20 (0.003)
9.85 (0.000)
12.13 (0.000)
10.21
Exc. 10 11.92 (0.034)
9.97 (0.004)
12.90 (0.004)
11.73
10.46 (0.014)
9.93 (0.003)
12.59 (0.004)
10.46
Exc. 11 12.03 (0.009)
10.15 (0.001)
13.07 (0.004)
11.94
10.49 (0.000)
10.08 (0.004)
12.78 (0.021)
10.50
Exc. 12 12.46 (0.000)
10.32 (0.010)
13.34 (0.013)
12.10
11.03 (0.030)
10.12 (0.000)
12.90 (0.004)
11.07
Exc. 13 12.91 (0.030)
10.39 (0.000)
13.37 (0.000)
12.87
11.13 (0.000)
10.15 (0.000)
13.07 (0.004)
11.14
Exc. 14 12.99 (0.064)
10.65 (0.011)
13.47 (0.007)
13.09
11.95 (0.013)
10.20 (0.002)
13.34 (0.013)
11.73
Exc. 15 13.16 (0.002)
11.14 (0.018)
13.53 (0.002)
13.52
12.35 (0.021)
10.29 (0.003)
13.36 (0.000)
12.09
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
8.14, 9.58
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
Exp.
8.14, 9.58
9.26, 9.63
9.26, 9.63
there is no need to use an upper limit of 5.
Similar calculations for testing optimal choice of lower and upper limit for neff have been
performed for acetone from which the same can be concluded. The calculations for acetone
are presented in Appendix A Tables A1-A3. To summon up on the facts the best choice of
the lower limit is 2.5 and the best choice of the upper limit is 4 or 5 depending on the user
to decide if it is necessary for the results. Some molecules are less effected than others but
the general tendency is that an upper limit of 4 is enough in most cases.
C.
Ionization potential
A center for the ionization potential is added to the calculations. The results are only
seen in some of the calculations because the ionization potentials lie higher in energy than
the first 15 excitations. Therefore the Hf-srDFT estimate of the ionization potentials are
13
TABLE IV. HF-srPBEGWS estimate of the ionization potentials (in eV) below 20 eV.
IP Formaldehyde Acetone Acetamide Thymine Cytosine Uracil
1
10.65
10.05
10.18
9.61
9.30
10.01
2
14.31
12.75
10.62
10.74
10.03
10.78
3
15.73
13.61
13.46
11.12
10.19
11.19
4
17.14
14.11
14.22
11.55
10.68
11.67
5
--
14.36
14.39
13.27
12.93
13.70
6
--
15.78
15.84
13.88
13.98
14.85
7
--
16.10
16.35
14.44
14.41
15.00
8
--
16.14
18.26
14.46
15.29
15.10
9
--
18.11
19.73
15.21
15.67
16.54
10
--
--
--
15.30
16.21
17.17
11
--
--
--
15.58
17.08
17.89
12
--
--
--
16.46
17.98
19.11
13
--
--
--
17.12
19.08
--
14
--
--
--
17.78
19.16
--
15
--
--
--
19.18
--
--
16
--
--
--
19.25
--
--
shown in Table 4. The energies are only shown for those less than 20 eV. The calculation
are based on the approximation that IP(HOMO + 1 − i) = −i where i is the orbital energy
of occupied orbital number i, also called Koopmans’ theorem //CITE-Koopmans//.
Comparing Table 4 and Table 5 which contain the description of IP in the basis set for
formaldehyde, then notice that the excited state 10.62 appearing between the excited states
10.39 and 10.65. The only difference is the IP basis function and thereby the excited state
10.62 must be an ionization potential which match the first calculated approximation for
formaldehyde in Table 4.
The ionization potential for acetone is easily seen, in Appendix A Table A3, with the
color code pink. The appearance of the extra excited state 10.02 between the excited states
9.90 and 10.19 similar to the first calculated ionization potential for acetone in Table 4.
The ionization energies in acetamine, cytosine, thymine and uracil is not present in the
14
TABLE V. The first 15 excited states for formaldehyde using the CAS-srPBEGWS method with
added Rydberg basis functions and an IP basis function. Excitation energies in eV and oscillator
strengths in brackets.
TZVP + Ryd-spd=2.5-4
TZVP + Ryd-spd=2.5-4 + IP
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
8.07 (0.000)
7.21 (0.018)
8.95 (0.001)
3.80
8.07 (0.000)
7.21 (0.018)
8.95 (0.001)
3.80
Exc. 2
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
Exc. 3
9.17 (0.018)
8.99 (0.015)
9.84 (0.000)
9.25
9.17 (0.018)
8.99 (0.015)
9.84 (0.000)
9.25
Exc. 4
9.44 (0.006)
9.13 (0.001) 10.48 (0.000) 9.50
9.44 (0.006)
9.13 (0.001) 10.48 (0.000) 9.50
Exc. 5
9.56 (0.110)
9.23 (0.001) 10.85 (0.043) 9.87
9.56 (0.110)
9.23 (0.001) 10.85 (0.043) 9.87
Exc. 6
9.93 (0.024)
9.41 (0.008) 11.65 (0.016) 9.91
9.93 (0.024)
9.41 (0.008) 11.65 (0.016) 9.92
Exc. 7
9.98 (0.007)
9.73 (0.008) 11.67 (0.010) 10.02 9.98 (0.007)
9.73 (0.008) 11.67 (0.010) 10.02
Exc. 8 10.55 (0.033) 9.81 (0.000) 12.59 (0.004) 10.54 10.55 (0.033) 9.81 (0.000) 12.59 (0.004) 10.54
Exc. 9 10.65 (0.003) 9.86 (0.000) 12.78 (0.020) 10.69 10.65 (0.003) 9.86 (0.000) 12.78 (0.020) 10.69
Exc. 10 11.92 (0.034) 9.97 (0.004) 12.90 (0.004) 11.73 11.93 (0.034) 9.97 (0.004) 12.90 (0.004) 11.73
Exc. 11 12.03 (0.009) 10.15 (0.001) 13.07 (0.004) 11.94 12.03 (0.009) 10.15 (0.001) 13.07 (0.004) 11.94
Exc. 12 12.46 (0.000) 10.32 (0.010) 13.34 (0.013) 12.10 12.46 (0.000) 10.32 (0.010) 13.34 (0.013) 12.10
Exc. 13 12.91 (0.030) 10.39 (0.000) 13.37 (0.000) 12.87 12.91 (0.030) 10.39 (0.000) 13.37 (0.000) 12.87
Exc. 14 12.99 (0.064) 10.65 (0.011) 13.47 (0.007) 13.09 12.99 (0.064) 10.62 (0.000) 13.47 (0.007) 13.09
Exc. 15 13.16 (0.002) 11.14 (0.018) 13.53 (0.002) 13.52 13.16 (0.002) 10.65 (0.011) 13.52 (0.002) 13.52
calculations because they are higher in energy than the first 15 excitations. They might be
seen if more excited states where calculated.
D.
Investigation of maximum ` for added Rydberg basis functions
To investigate the choice of the largest number of angular quantum numbers. The idea
is to see all the Rydberg basis functions that will appear when calculating max(`) = 0, 1,
2, 3, 4. Then to be able to see the Rydberg states closing up on the ionization potential the
molecule acetone is used as an example. Recall that the first ionization potential of acetone
is 10.02 eV as discussed earlier in section 4.3.
Compare the “TZVP + Ryd-s=2.5-5” (left side of Table 6) with the upgrade “TZVP +
Ryd-sp=2.5-5” (right side of Table 6). To show how the Rydberg states are appearing, the
15
excited states with oscillator strength above 0.100 (valence excitations) have been marked
green which makes it easier to see the change. Notice in “TZVP + Ryd-s=2.5-5” (left side
of Table 6) the first excited state in symmetry A1 is 9.11 eV with oscillator strength 0.250
then the next valence state is in the third excited state in symmetry A1 at 10.94 eV with
oscillator strength 0.245. When observing the same two states in “TZVP + Ryd-sp=2.5-5”
(right side of Table 6) see that two Rydberg states are appearing as excited states 1 and
2. Also between the two previous states there are now appearing four additional Rydberg
states. Addional Ryberg states are similarly seen for “TZVP + Ryd-sp=2.5-5” and to “TZVP
+ Ryd-spd=2.5-5” see table C1 and again for spd to spdf in C2 in Appendix C.
The additional Rydberg states are excellent in the case of Rydberg-valence mixing, but
the upgrade to “TZVP + Ryd-spdf=2.5-5” and “TZVP + Ryd-spdfg=2.5-5” results in an
overload of Rydberg states which is not necessary for this investigation. The idea is to get a
description, which leads to the best match to the experimental data. The excess of Rydberg
states is not helping in this matter.
Recall that the first ionization potential for acetone is 10.02 and notice that the higher
value of maximum ` the more Rydberg states are showing near and up to 10.02 eV. This
confirms the theory about the unlimited number of Rydberg states towards the ionization
potential at 10.02 eV. To see the “TZVP + Ryd-spdfg=2.5-5” (right side of Table 7) which
is the largest calculated basis set with the most Rydberg states closing up to the ionization
potential at 10.02 eV for acetone.
E.
The influence of MC-srDFT
As already concluded the calculations provide good estimates for the excitation energies.
Then it is of interest to check the influence of MC-srDFT in the calculations. First step is to
show that the multiconfigurational part of the method has influence on the energies. This is
done by neglecting the MC-srDFT part and run HF-srDFT with the short-range PBEGWS
functional instead. To illustrate this the molecule formaldehyde is shown as an example in
Table 8.
The left side of Table 8 show the excited states for the CAS-srPBEGWS method and
the right side of Table 8 show the excited states for the HF-srPBEGWS method. Notice
that the HF-srPBE method does not perform as satisfactory as the CAS-srPBE method.
16
TABLE VI. The first 25 excited states for acetone using CAS-srPBEGWS with two different `
ranges: “s” and “sp”. Excitation energies in eV and oscillator strengths in parentheses.
TZVP + Ryd-s=2.5-5
TZVP + Ryd-sp=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
9.11 (0.215)
7.02 (0.028)
8.91 (0.000)
4.31
7.97 (0.002)
7.02 (0.034)
8.90 (0.000)
4.30
Exc. 2 10.18 (0.082)
8.60 (0.000)
9.65 (0.013)
8.67
8.95 (0.058)
7.93 (0.000)
9.61 (0.016)
7.84
Exc. 3 10.94 (0.245)
9.20 (0.001)
11.09 (0.023)
9.82
9.17 (0.198)
8.61 (0.000)
10.61 (0.052)
8.66
Exc. 4 10.04 (0.018)
9.49 (0.002)
11.23 (0.006)
10.20
9.42 (0.013)
8.98 (0.000)
11.05 (0.022)
8.94
Exc. 5 11.95 (0.001)
9.66 (0.003)
11.28 (0.008)
10.97
9.64 (0.002)
9.22 (0.001)
11.23 (0.006)
9.38
Exc. 6 12.29 (0.040)
9.79 (0.008)
11.88 (0.003)
11.36
9.96 (0.001)
9.40 (0.000)
11.27 (0.006)
9.62
Exc. 7 12.47 (0.119)
9.97 (0.003)
12.17 (0.004)
12.50
10.51 (0.020)
9.51 (0.001)
11.66 (0.038)
9.81
Exc. 8 12.67 (0.023) 10.24 (0.013) 12.33 (0.007)
12.63
10.55 (0.044)
9.63 (0.000)
11.87 (0.015)
9.92
Exc. 9 12.79 (0.058) 10.37 (0.009) 12.44 (0.000)
12.82
10.80 (0.138)
9.67 (0.004)
11.91 (0.027)
10.41
Exc. 10 13.25 (0.003) 10.57 (0.103) 12.47 (0.448)
13.01
11.05 (0.097)
9.81 (0.006)
11.98 (0.054)
10.51
Exc. 11 13.42 (0.011) 10.87 (0.164) 12.60 (0.000)
13.16
11.46 (0.003)
9.95 (0.000)
12.10 (0.001)
10.66
Exc. 12 13.54 (0.001) 12.09 (0.019) 12.97 (0.002)
13.41
11.63 (0.000)
9.99 (0.005)
12.20 (0.009)
10.98
Exc. 13 13.68 (0.006) 12.21 (0.035) 13.14 (0.014)
13.71
11.96 (0.001) 10.27 (0.005) 12.33 (0.005)
11.31
Exc. 14 13.78 (0.026) 12.68 (0.095) 13.26 (0.018)
13.90
12.02 (0.001) 10.31 (0.052) 12.36 (0.007)
11.49
Exc. 15 13.91 (0.027) 12.74 (0.056) 13.48 (0.026)
14.04
12.07 (0.001) 10.65 (0.152) 12.47 (0.004)
11.69
Exc. 16 14.10 (0.004) 12.82 (0.103) 13.62 (0.077)
14.04
12.31 (0.002) 10.89 (0.087) 12.58 (0.004)
12.09
Exc. 17 14.37 (0.001) 12.97 (0.076) 14.05 (0.032)
14.32
12.38 (0.016) 11.70 (0.020) 12.65 (0.014)
12.32
Exc. 18 14.39 (0.069) 13.12 (0.097) 14.38 (0.036)
14.65
12.60 (0.024) 11.83 (0.001) 12.94 (0.072)
12.48
Exc. 19 14.49 (0.002) 13.25 (0.066) 14.46 (0.000)
14.92
12.61 (0.020) 11.97 (0.043) 12.99 (0.000)
12.54
Exc. 20 14.94 (0.000) 13.40 (0.036) 14.63 (0.033)
15.18
12.69 (0.053) 12.08 (0.003) 13.05 (0.021)
12.65
Exc. 21 15.04 (0.075) 13.43 (0.039) 14.75 (0.003)
15.46
12.83 (0.006) 12.22 (0.003) 13.18 (0.010)
12.74
Exc. 22 15.13 (0.003) 13.59 (0.006) 14.93 (0.032)
15.71
12.98 (0.005) 12.60 (0.003) 13.31 (0.028)
12.93
Exc. 23 15.24 (0.000) 13.94 (0.003) 15.20 (0.067)
15.99
13.10 (0.010) 12.68 (0.061) 13.37 (0.000)
13.06
Exc. 24 15.37 (0.010) 14.22 (0.035) 15.39 (0.149)
16.11
13.18 (0.021) 12.76 (0.105) 13.44 (0.005)
13.16
Exc. 25 15.40 (0.000) 14.32 (0.114) 15.48 (0.001)
16.15
13.21 (0.007) 12.86 (0.056) 13.50 (0.032)
13.30
Exp.
7.41, 7.80
6.36, 7.49, 8.09
8.17
4.43, 7.36
7.41, 7.80
6.36, 7.49, 8.09
8.17
4.43, 7.36
To illustrate the excitations for CAS-srPBE that fits the experimental data is colored blue.
The corresponding excited states for the HF-srPBE is colored green and yellow, accordingly
if the energies are less than 0.03 eV from the CAS-srPBE energies and yellow if they are
more than 0.03 eV from the CAS-srPBE energies.
The difference here is not huge but the general tendency is that the CAS-srPBE is a
17
TABLE VII. The first 25 excited states for acetone using CAS-srPBEGWS with two different `
ranges: “spd” and “spdf ”. Excitation energies in eV and oscillator strengths in parentheses.
TZVP + Ryd-spdf=2.5-5
TZVP + Ryd-spdfg=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
7.97 (0.006)
7.00 (0.029)
8.63 (0.007)
4.29
7.96 (0.006)
7.00 (0.029)
8.63 (0.007)
4.29
Exc. 2
8.57 (0.038)
7.93 (0.001)
8.89 (0.000)
7.84
8.57 (0.038)
7.93 (0.001)
8.89 (0.000)
7.84
Exc. 3
8.94 (0.085)
8.37 (0.016)
9.19 (0.000)
8.55
8.94 (0.087)
8.36 (0.017)
9.19 (0.000)
8.55
Exc. 4
9.13 (0.145)
8.60 (0.003)
9.26 (0.003)
8.66
9.13 (0.143)
8.60 (0.003)
9.26 (0.003)
8.65
Exc. 5
9.21 (0.007)
8.72 (0.009)
9.51 (0.000)
8.95
9.20 (0.007)
8.72 (0.009)
9.51 (0.000)
8.95
Exc. 6
9.22 (0.022)
8.99 (0.001)
9.55 (0.003)
9.18
9.22 (0.021)
8.99 (0.001)
9.53 (0.000)
9.18
Exc. 7
9.23 (0.021)
9.12 (0.001)
9.58 (0.013)
9.22
9.23 (0.020)
9.12 (0.001)
9.53 (0.000)
9.22
Exc. 8
9.43 (0.011)
9.16 (0.005)
9.78 (0.000)
9.23
9.43 (0.011)
9.16 (0.006)
9.55 (0.003)
9.23
Exc. 9
9.52 (0.001)
9.22 (0.000)
9.86 (0.002)
9.39
9.52 (0.001)
9.21 (0.000)
9.58 (0.013)
9.39
Exc. 10 9.53 (0.002)
9.25 (0.001)
10.22 (0.000)
9.50
9.52 (0.000)
9.25 (0.001)
9.77 (0.000)
9.50
Exc. 11 9.54 (0.009)
9.29 (0.004)
10.47 (0.005)
9.53
9.53 (0.000)
9.29 (0.004)
9.79 (0.000)
9.53
Exc. 12 9.66 (0.004)
9.41 (0.000)
10.55 (0.055)
9.54
9.53 (0.002)
9.41 (0.000)
9.80 (0.000)
9.53
Exc. 13 9.80 (0.002)
9.46 (0.000)
10.94 (0.003)
9.63
9.54 (0.009)
9.45 (0.000)
9.85 (0.002)
9.53
Exc. 14 9.82 (0.000)
9.49 (0.004)
10.98 (0.001)
9.77
9.64 (0.004)
9.49 (0.004)
10.20 (0.001)
9.54
Exc. 15 9.82 (0.011)
9.53 (0.000)
11.02 (0.017)
9.80
9.79 (0.000)
9.52 (0.000)
10.26 (0.000)
9.63
Exc. 16 9.98 (0.010)
9.55 (0.001)
11.24 (0.008)
9.83
9.79 (0.000)
9.53 (0.000)
10.26 (0.000)
9.77
Exc. 17 10.29 (0.001)
9.57 (0.003)
11.37 (0.007)
9.84
9.80 (0.001)
9.53 (0.000)
10.47 (0.005)
9.79
Exc. 18 10.34 (0.003)
9.64 (0.000)
11.63 (0.052)
9.93
9.82 (0.001)
9.53 (0.000)
10.55 (0.054)
9.80
Exc. 19 10.39 (0.021)
9.70 (0.000)
11.69 (0.000)
10.24
9.82 (0.011)
9.55 (0.001)
10.91 (0.004)
9.80
Exc. 20 10.48 (0.025)
9.70 (0.001)
11.75 (0.003)
10.36
9.98 (0.010)
9.57 (0.003)
10.95 (0.000)
9.83
Exc. 21 10.73 (0.108)
9.79 (0.006)
11.79 (0.000)
10.37
10.24 (0.000)
9.64 (0.000)
11.01 (0.015)
9.83
Exc. 22 10.77 (0.004)
9.82 (0.001)
11.81 (0.005)
10.41
10.25 (0.000)
9.69 (0.000)
11.08 (0.000)
9.93
Exc. 23 11.21 (0.000)
9.85 (0.001)
11.90 (0.011)
10.54
10.29 (0.001)
9.70 (0.001)
11.10 (0.001)
10.23
Exc. 24 11.24 (0.000)
9.95 (0.001)
11.91 (0.010)
10.90
10.34 (0.003)
9.79 (0.000)
11.24 (0.008)
10.25
Exc. 25 11.31 (0.013)
9.96 (0.000)
11.94 (0.010)
11.14
10.39 (0.020)
9.79 (0.003)
11.37 (0.007)
10.26
6.36, 7.49, 8.09
8.17
4.43, 7.36
7.41, 7.80
6.36, 7.49, 8.09
8.17
4.43, 7.36
Exp.
7.41, 7.80
better estimate, though three of the energies from HF-srDFT have an estimate closer to
the experimental data. The first excited state in symmetry A1 at 8.13, excited state 5 in
symmetry B1 at 9.24 and excited state 3 in symmetry A2 at 9.24. Compared to the CASsrPBE where six of the excited states are a better match to the experimental data. Though
the difference between the two methods are not significant. The MC part have a positive
18
TABLE VIII. The first 15 excited states for formaldehyde using the MC-srDFT methods CASsrPBEGWS and HF-srPBEGWS. Excitation energies in eV and oscillator strengths in parentheses.
CAS-srPBEGWS
HF-srPBEGWS
TZVP + Ryd-spd=2.5-5
TZVP + Ryd-spd=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
8.07 (0.000)
7.20 (0.018)
8.95 (0.001)
3.80
8.13 (0.042)
7.28 (0.012)
9.05 (0.001)
3.83
Exc. 2
8.10 (0.043)
7.96 (0.038)
9.18 (0.000)
8.28
9.16 (0.017)
8.02 (0.043)
9.18 (0.000)
8.29
Exc. 3
9.17 (0.018)
8.98 (0.014)
9.83 (0.000)
9.25
9.45 (0.012)
9.00 (0.011)
9.82 (0.000)
9.24
Exc. 4
9.44 (0.006)
9.13 (0.001)
10.14 (0.000)
9.50
9.56 (0.112)
9.12 (0.003)
10.13 (0.000)
9.49
Exc. 5
9.56 (0.110)
9.22 (0.001)
10.42 (0.000)
9.86
9.89 (0.020)
9.24 (0.001)
10.41 (0.000)
9.85
Exc. 6
9.90 (0.018)
9.41 (0.008)
10.84 (0.044)
9.91
9.94 (0.005)
9.42 (0.010)
10.90 (0.051)
9.96
Exc. 7
9.95 (0.008)
9.73 (0.007)
11.00 (0.000)
9.97
10.17 (0.008)
9.74 (0.006)
11.00 (0.000)
10.14
Exc. 8 10.17 (0.008)
9.81 (0.000)
11.65 (0.026)
10.16
10.19 (0.003)
9.79 (0.001)
11.69 (0.018)
10.18
Exc. 9 10.20 (0.003)
9.85 (0.000)
12.13 (0.000)
10.21
10.46 (0.013)
9.85 (0.000)
12.12 (0.000)
10.21
Exc. 10 10.46 (0.014)
9.93 (0.003)
12.59 (0.004)
10.46
10.49 (0.000)
9.93 (0.004)
12.61 (0.002)
10.45
Exc. 11 10.49 (0.000)
10.08 (0.004)
12.78 (0.021)
10.50
11.04 (0.028)
10.08 (0.003)
12.75 (0.022)
10.50
Exc. 12 11.03 (0.030)
10.12 (0.000)
12.90 (0.004)
11.07
11.12 (0.000)
10.11 (0.001)
12.89 (0.005)
11.06
Exc. 13 11.13 (0.000)
10.15 (0.000)
13.07 (0.004)
11.14
11.94 (0.014)
10.14 (0.000)
13.07 (0.003)
11.13
Exc. 14 11.95 (0.013)
10.20 (0.002)
13.34 (0.013)
11.73
12.39 (0.028)
10.19 (0.003)
13.35 (0.011)
11.75
Exc. 15 12.35 (0.021)
10.29 (0.003)
13.36 (0.000)
12.09
12.57 (0.002)
10.29 (0.003)
13.37 (0.002)
12.41
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
8.14, 9.58
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
Exp.
8.14, 9.58
9.26, 9.63
9.26, 9.63
impact on the energies.
As mentioned in the theory section, the whole idea of MC-srDFT is the parameter.
Therefore the parameter is set to 0.01 which results in the KS-DFT method. The PBE
method(left side of Table 9) are compared to the results from the CAS-srPBE(left side of
Table 8) to make sure that CAS-srPBE is the better method in this case.
To check the method of CASSCF(right side of Table 9) compared to the CAS-srPBE(left
side of Table 8) the value is set to 50. The results of both methods clearly show that non
of the excited states match the experimental result as satisfactory as CAS-srPBE. The PBE
method is far from the experimental data and is clearly a poorer choice of method in this
case. The CASSCF are a better choice compared to PBE but again far from the results
accomplished by the CAS-srPBE.
This clearly demonstrate that the CAS-srPBE performs better in this matter than both
PBE and CASSCF. Therefore, as a result of this investigation, it can be concluded that
19
TABLE IX. The first 15 excited states for formaldehyde using pure DFT KS-DFT(PBE) and pure
wave function theory CASSCF. Excitation energies in eV and oscillator strengths in parentheses.
PBE
CASSCF
TZVP + Ryd-spd=2.5-5
TZVP + Ryd-spd=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
A2
Exc. 1
6.18 (0.000)
5.80 (0.020)
6.25 (0.000)
3.74
8.49 (0.052)
7.47 (0.009)
9.56 (0.002)
4.32
Exc. 2
6.25 (0.000)
6.15 (0.000)
6.48 (0.000)
6.18
9.42 (0.084)
8.23 (0.057)
9.59 (0.000)
8.68
Exc. 3
6.29 (0.003)
6.18 (0.000)
6.85 (0.000)
6.25
9.80 (0.067)
9.45 (0.016)
10.24 (0.000)
9.67
Exc. 4
6.46 (0.005)
6.24 (0.002)
7.43 (0.000)
6.31
9.85 (0.024)
9.56 (0.001)
10.55 (0.000)
9.91
Exc. 5
6.49 (0.002)
6.25 (0.000)
8.47 (0.000)
6.48
10.30 (0.010)
9.62 (0.000)
10.83 (0.000)
10.28
Exc. 6
6.75 (0.012)
6.26 (0.000)
8.75 (0.001)
6.52
10.36 (0.008)
9.78 (0.011)
11.38 (0.052)
10.38
Exc. 7
6.86 (0.001)
6.31 (0.001)
9.56 (0.015)
6.83
10.58 (0.005)
10.17 (0.008)
11.41 (0.001)
10.57
Exc. 8
7.33 (0.009)
6.39 (0.002)
9.89 (0.000)
6.88
10.61 (0.004)
10.22 (0.000)
12.29 (0.028)
10.62
Exc. 9
7.48 (0.003)
6.47 (0.000)
9.92 (0.000)
7.44
10.87 (0.013)
10.26 (0.000)
12.59 (0.000)
10.86
Exc. 10
8.34 (0.000)
6.48 (0.000)
9.98 (0.000)
7.51
10.90 (0.000)
10.33 (0.004)
13.23 (0.009)
10.91
Exc. 11
8.34 (0.003)
6.51 (0.002)
10.00 (0.000)
8.51
11.44 (0.033)
10.50 (0.004)
13.42 (0.027)
11.05
Exc. 12
8.60 (0.020)
6.69 (0.010)
10.00 (0.000)
8.73
11.52 (0.000)
10.54 (0.000)
13.54 (0.002)
11.50
Exc. 13
9.36 (0.101)
6.80 (0.001)
10.05 (0.002)
9.61
12.46 (0.017)
10.56 (0.000)
13.71 (0.005)
11.55
Exc. 14
9.92 (0.000)
6.85 (0.000)
10.06 (0.000)
9.92
12.84 (0.028)
10.60 (0.003)
14.00 (0.005)
12.31
Exc. 15 10.00 (0.000)
6.87 (0.001)
10.13 (0.000)
10.00
13.57 (0.004)
10.70 (0.003)
14.10 (0.009)
12.87
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
8.14, 9.58
7.08, 7.97, 8.88,
9.00
4.00, 8.37, 9.22
Exp.
8.14, 9.58
9.26, 9.63
9.26, 9.63
the satisfactory result are due to the MC-srDFT method and not just the Rydberg basis
functions added to the basis set.
These conclusions are in some way similar to the conclusions from (Hubert, Hedegård,
& Jensen, 2016) where it was noted that the mixing errors were larger with KS-DFT and
almost similar with HF-srDFT. Though the research from the paper, in contrary found that
the CASSCF were improving. In this research paper the MC-srDFT is concluded to be the
method that improves the results.
V.
CONCLUSIONS
The purpose of this investigation was to see how the MC-srDFT method was performing
with a proper basis set description for the Rydberg states in order to avoid spurious Rydbergvalence mixing. This has been done by adding center of mass Rydberg basis functions to the
basis set according to the recipe by Kaufmann et al. (Karl Kaufmann, 1989). Furthermore
20
the investigation should suggest a recipe of how to choose a proper basis set, to get the best
description of the excited states of molecules with Rydberg-valence mixing. The detailed
results have been discussed in section 4.
The MC-srDFT method together with the basis set containing the center of mass function
was a success. The resulting excited states had a very satisfactory match to the experimental
data. One notices that the Rydberg states appearing between the valence excitations. This
means that there now is a satisfactory description for Rydberg and valence and therefore less
spurious Rydberg-valence mixing. Then to get the best results the Rydberg basis functions
were adjusted and investigated with different values. The result here was to set the minimum
and maximum neff to be accordingly 2.5 and 4. By adding the IP to the basis set it is possible
to get a good description of the ionization potentials in the calculations. To investigate the
effect of restricting maximum angular quantum number ` the different values was tried
out with s=0, p=1, d=2, etc. A higher maximum will obviously give a lot more Rydberg
states than necessary for this purpose and too low a maximum will result in a poorer result.
Therefore the maximum ` is set to 2. With this recipe for the Rydberg basis set of molecules
with Rydberg-valence mixing, one has a good chance of getting satisfactory results using
the MC-srDFT method.
To make sure that the satisfactory results is thanks to the MC-srDFT method, the KSDFT and MCSCF methods are applied to the same basis set. The result was not near as
good as those of the MC-srDFT method. Hence the conclusion that the MC-srDFT is the
reason for these fine results.
VI.
REFERENCES
(1966). In G. Herzberg, Molecular spectra and molecular structure (pp. 411-417). Van
Nostrand Reinhold Company.
Atkins, P., & Friedman, R. (2005). Molecular quantum mechanics. Oxford University
Press.
Buenker, R. J., Hirsch, G., & Li, Y. (n.d.). Ab Initio configuration interaktion calcuations
of rydberg and mixed valens-ryberg states .
Dalton Program manual. (2015).
Fromager, E., Toulouse, J., & Jensen, H. A. (2007). On the universality of the long-
1
/short-range separation in multiconfigurational density-functional theory. J. Chem. Phys.
126, 074111.
Hubert, M., Hedegård, E. D., & Jensen, H. J. (2016, April 8). Investigation of Multiconfigurational short-range Density Functional Theory for Electronic Excitations in Organic
Molecules. Journal of Chemical Theory and Computation.
Karl Kaufmann, W. B. (1989, March 22). Universal Gausian basis set for an optimum
representation of Rydberg and continuum wavefunctions. Institute of Physical Chemitry,
University of Basel, Switzerland.
Kenneth B. Wiberg, A. E. (2002, February 8). A Comparison of the Electronic Transition
Energies for Ethene, Isobutene, Formaldehyde and Acetone Calculated Using RPA, TDDFT
and EOM-CCSD. Effect of Basis Sets. pp. 4192-4199.
Koopmans’ theorem. (n.d.). Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Koopmans%27_
Rydberg transitions. (2000). Retrieved from http://vergil.chemistry.gatech.edu/notes/excd/node2.html
2
TABLE X. Table A1: The first 15 excited states for acetone using CAS-srPBEGWS without and
with added Rydberg basis functions. Excitation energies in eV and oscillator strengths in parentheses. Experimental data in last row.
TZVP
TZVP + Ryd-spd=2.5-5
Sym.
A1
B1
B2
A2
Exc. 1
9.12 (0.214)
8.11 (0.029)
8.91 (0.000)
4.31
7.97 (0.006) 7.00 (0.030) 8.63 (0.007)
4.29
Exc. 2 10.19 (0.071) 9.82 (0.024) 10.61 (0.048)
8.68
8.57 (0.033) 7.93 (0.000) 8.90 (0.000)
7.84
Exc. 3 10.96 (0.149) 10.34 (0.022) 11.09 (0.030)
9.82
8.95 (0.070) 8.37 (0.016) 9.26 (0.003)
8.56
Exc. 4 11.79 (0.210) 10.73 (0.147) 11.23 (0.009)
10.20
9.16 (0.023) 8.61 (0.004) 9.55 (0.003)
8.66
Exc. 5 11.98 (0.026) 10.95 (0.096) 12.40 (0.000)
11.36
9.23 (0.042) 8.73 (0.011) 9.58 (0.013)
8.95
Exc. 6 12.47 (0.121) 11.33 (0.111) 12.82 (0.221)
11.49
9.43 (0.014) 8.99 (0.000) 9.85 (0.002)
9.22
Exc. 7 12.75 (0.075) 12.61 (0.028) 13.04 (0.175)
12.49
9.54 (0.010) 9.13 (0.005) 10.45 (0.002)
9.39
Exc. 8 12.86 (0.001) 12.89 (0.431) 13.16 (0.000)
12.74
9.66 (0.005) 9.25 (0.002) 10.56 (0.057)
9.53
Exc. 9 13.68 (0.025) 13.42 (0.030) 13.44 (0.043)
13.01
9.82 (0.010) 9.29 (0.004) 10.97 (0.004)
9.63
Exc. 10 13.84 (0.013) 13.45 (0.049) 13.60 (0.257)
13.18
9.98 (0.010) 9.41 (0.000) 11.03 (0.020)
9.80
Exc. 11 14.35 (0.088) 13.54 (0.055) 13.84 (0.002)
13.88
10.39 (0.023) 9.48 (0.002) 11.07 (0.000)
9.81
Exc. 12 14.38 (0.006) 14.28 (0.118) 14.40 (0.006)
14.10
10.48 (0.037) 9.55 (0.000) 11.28 (0.006)
9.93
Exc. 13 14.90 (0.080) 14.34 (0.001) 14.47 (0.000)
14.85
10.74 (0.110) 9.56 (0.023) 11.38 (0.008)
10.34
Exc. 14 15.14 (0.008) 14.67 (0.111) 14.55 (0.006)
15.09
10.75 (0.004) 9.64 (0.000) 11.64 (0.046)
10.38
Exc. 15 15.41 (0.011) 14.81 (0.091) 14.76 (0.001)
15.40
11.27 (0.000) 9.70 (0.000) 11.73 (0.006)
10.62
Exp.
7.41, 7.80
6.36, 7.49,
8.17
4.43, 7.36
A1
7.41, 7.80
8.09
Blue - The excited states close to the experimental data.
Green small change from 0.03-0.06 eV
Blue change from 0.07-0.10 eV
Yellow change more than 0.10 eV
Pink - The ionization potential
B1
6.36, 7.49,
8.09
B2
8.17
A2
4.43, 7.36
3
TABLE XI. Table A2: The first 15 excited states for acetone using CAS-srPBEGWS with added
Rydberg basis functions with different lower limits for neff . Excitation energies in eV and oscillator
strengths in parentheses.
TZVP + Ryd-spd=3-4
TZVP + Ryd-spd=2.5-4
Sym.
A1
B1
B2
A2
A1
B1
B2
Exc. 1
7.98 (0.004)
7.12 (0.029)
8.64 (0.005)
4.30
7.96 (0.006)
7.00 (0.030)
8.63 (0.007)
Exc. 2
8.58 (0.025)
7.96 (0.000)
8.91 (0.000)
7.91
8.57 (0.033)
7.93 (0.001)
8.90 (0.000)
Exc. 3
8.97 (0.048)
8.44 (0.018)
9.29 (0.003)
8.58
8.95 (0.069)
8.37 (0.017)
9.28 (0.003)
Exc. 4
9.19 (0.189)
8.61 (0.003)
9.70 (0.018)
8.67
9.16 (0.189)
8.61 (0.004)
9.59 (0.014)
Exc. 5
9.27 (0.037)
8.78 (0.007)
9.91 (0.004)
8.99
9.25 (0.022)
8.73 (0.011)
9.92 (0.004)
Exc. 6
9.51 (0.020)
9.00 (0.000) 10.61 (0.046)
9.25
9.52 (0.021)
8.99 (0.000) 10.56 (0.056)
Exc. 7
9.87 (0.032)
9.17 (0.006) 11.04 (0.015)
9.46
9.86 (0.028)
9.13 (0.005) 10.91 (0.002)
Exc. 8 10.56 (0.024) 9.28 (0.002) 11.07 (0.010)
9.81
10.32 (0.016) 9.27 (0.002) 10.99 (0.000)
Exc. 9 10.81 (0.108) 9.33 (0.005) 11.24 (0.008)
9.83
10.51 (0.016) 9.30 (0.005) 11.04 (0.019)
Exc. 10 11.28 (0.000) 9.48 (0.001) 11.29 (0.004)
10.43
10.74 (0.107) 9.48 (0.000) 11.28 (0.006)
Exc. 11 11.39 (0.019) 9.65 (0.003) 11.44 (0.010)
10.97
11.16 (0.058) 9.57 (0.000) 11.38 (0.009)
Exc. 12 11.65 (0.005) 9.89 (0.005) 11.68 (0.033)
11.27
11.27 (0.003) 9.74 (0.004) 11.63 (0.063)
Exc. 13 11.68 (0.000) 10.09 (0.003) 11.79 (0.013)
11.30
11.36 (0.000) 9.90 (0.007) 11.66 (0.000)
Exc. 14 11.82 (0.006) 10.32 (0.009) 11.82 (0.001)
11.41
11.63 (0.001) 10.19 (0.002) 11.76 (0.001)
Exc. 15 11.90 (0.008) 10.38 (0.068) 11.93 (0.028)
11.54
11.84 (0.005) 10.24 (0.000) 11.79 (0.004)
Exp.
7.41, 7.80
6.36, 7.49,
8.09
8.17
4.43, 7.36
7.41, 7.80
6.36, 7.49,
8.09
8.17
A2
4.43, 7.36
4
TABLE XII. Table A3: The first 15 excited states for acetone using CAS-srPBEGWS and added
Rydberg basis functions in the basis set with different upper limits for neff . Excitation energies in
eV and oscillator strengths in brackets.
TZVP + Ryd-spd=2.5-4
TZVP + Ryd-spd=2.5-4 + IP
Sym.
A1
B1
B2
A2
A1
B1
B2
Exc. 1
7.96 (0.006)
7.00 (0.030)
8.63 (0.007)
4.30
7.96 (0.006)
7.00 (0.030)
8.63 (0.007)
Exc. 2
8.57 (0.033)
7.93 (0.001)
8.90 (0.000)
7.84
8.57 (0.033)
7.93 (0.001)
8.90 (0.000)
Exc. 3
8.95 (0.069)
8.37 (0.017)
9.28 (0.003)
8.56
8.95 (0.069)
8.37 (0.017)
9.28 (0.003)
Exc. 4
9.16 (0.189)
8.61 (0.004)
9.59 (0.014)
8.66
9.16 (0.189)
8.61 (0.004)
9.59 (0.014)
Exc. 5
9.25 (0.022)
8.73 (0.011)
9.92 (0.004)
8.95
9.25 (0.022)
8.73 (0.011)
9.92 (0.004)
Exc. 6
9.52 (0.021)
8.99 (0.000) 10.56 (0.056)
9.23
9.52 (0.021)
8.99 (0.000) 10.56 (0.056)
Exc. 7
9.86 (0.028)
9.13 (0.005) 10.91 (0.002)
9.45
9.86 (0.028)
9.13 (0.005) 10.91 (0.002)
Exc. 8 10.32 (0.016) 9.27 (0.002) 10.99 (0.000)
9.80
10.32 (0.016) 9.27 (0.002) 10.99 (0.000)
Exc. 9 10.51 (0.016) 9.30 (0.005) 11.04 (0.019)
9.83
10.51 (0.016) 9.30 (0.005) 11.04 (0.019)
Exc. 10 10.74 (0.107) 9.48 (0.000) 11.28 (0.006)
10.18
10.74 (0.107) 9.48 (0.000) 11.28 (0.006)
Exc. 11 11.16 (0.058) 9.57 (0.000) 11.38 (0.009)
10.37
11.16 (0.058) 9.57 (0.000) 11.38 (0.008)
Exc. 12 11.27 (0.003) 9.74 (0.004) 11.63 (0.063)
10.91
11.27 (0.033) 9.74 (0.004) 11.63 (0.063)
Exc. 13 11.36 (0.000) 9.90 (0.007) 11.66 (0.000)
10.98
11.36 (0.000) 9.90 (0.007) 11.66 (0.000)
Exc. 14 11.63 (0.001) 10.19 (0.002) 11.76 (0.001)
11.23
11.63 (0.001) 10.02 (0.000) 11.76 (0.001)
Exc. 15 11.84 (0.005) 10.24 (0.000) 11.79 (0.004)
11.27
11.84 (0.005) 10.19 (0.002) 11.79 (0.004)
Exp.
7.41, 7.80
6.36, 7.49,
8.09
8.17
4.43, 7.36
7.41, 7.80
6.36, 7.49,
8.09
8.17
A2
4.43, 7.36
5
TABLE XIII. Table B1: The first 15 excited states for acetamide using CAS-srPBEGWS without
and with added Rydberg basis functions and IP. Excitation energies in eV and oscillator strengths
in parentheses. Experimental data in last row.
TZVP
Sym.
A0
TZVP+Ryd-spd=2.5-5
A00
A0
A00
TZVP+Ryd-spd=2.5-4 TZVP+Ryd-spd=2.5-4+IP
A0
A00
A0
Exc. 1 7.65 (0.0086) 5.58 (0.001) 6.93 (0.025) 5.53 (0.001) 6.93 (0.025) 5.53 (0.000) 6.93 (0.025)
Exc. 2
8.09 (0.156)
7.75 (0.010) 7.47 (0.159) 6.99 (0.031) 7.47 (0.159) 6.99 (0.015) 7.46 (0.160)
Exc. 3
9.29 (0.045)
9.05 (0.000) 7.86 (0.006) 7.96 (0.000) 7.86 (0.005) 7.96 (0.000) 7.86 (0.005)
Exc. 4
9.65 (0.040)
9.83 (0.000) 7.95 (0.040) 8.10 (0.000) 7.95 (0.040) 8.10 (0.000) 7.95 (0.040)
Exc. 5 10.20 (0.050) 10.00 (0.001) 8.48 (0.008) 8.37 (0.004) 8.48 (0.008) 8.37 (0.004) 8.48 (0.008)
Exc. 6 10.39 (0.157) 10.71 (0.000) 8.53 (0.026) 8.67 (0.000) 8.53 (0.026) 8.67 (0.000) 8.53 (0.026)
Exc. 7 10.84 (0.119) 10.73 (0.015) 8.61 (0.001) 8.70 (0.006) 8.61 (0.001) 8.70 (0.006) 8.61 (0.001)
Exc. 8 11.50 (0.001) 10.88 (0.006) 8.63 (0.027) 9.02 (0.000) 8.63 (0.027) 9.02 (0.000) 8.63 (0.027)
Exc. 9 11.63 (0.019) 11.16 (0.004) 8.74 (0.023) 9.03 (0.000) 8.74 (0.023) 9.03 (0.000) 8.76 (0.023)
Exc. 10 11.71 (0.241) 11.50 (0.031) 9.01 (0.001) 9.11 (0.000) 9.01 (0.001) 9.11 (0.000) 9.01 (0.000)
Exc. 11 12.09 (0.120) 12.04 (0.055) 9.02 (0.006) 9.22 (0.008) 9.02 (0.006) 9.22 (0.008) 9.02 (0.005)
Exc. 12 12.38 (0.070) 12.11 (0.033) 9.21 (0.002) 9.26 (0.001) 9.21 (0.002) 9.26 (0.001) 9.21 (0.002)
Exc. 13 12.61 (0.066) 12.37 (0.041) 9.25 (0.011) 9.31 (0.000) 9.25 (0.011) 9.31 (0.000) 9.24 (0.010)
Exc. 14 13.10 (0.021) 12.56 (0.000) 9.27 (0.000) 9.34 (0.003) 9.27 (0.000) 9.34 (0.003) 9.27 (0.000)
Exc. 15 13.31 (0.038) 12.62 (0.015) 9.30 (0.004) 9.48 (0.000) 9.30 (0.004) 9.48 (0.000) 9.28 (0.004)
Exp.
A00
6
TABLE XIV. Table B2: The first 15 excited states for cytosine using the CAS-srPBEGWS method
without and with added Rydberg basis functions and IP. Excitation energies in eV and oscillator
strengths in parentheses. Experimental data in last row.
TZVP
Sym.
A0
TZVP+Ryd-spd=2.5-5
A00
A0
A00
TZVP+Ryd-spd=2.5-4 TZVP+Ryd-spd=2.5-4+IP
A0
A00
A0
Exc. 1
4.93 (0.061) 5.06 (0.002) 4.91 (0.061) 5.05 (0.002) 4.91 (0.061) 5.05 (0.002) 4.91 (0.061)
Exc. 2
5.95 (0.128) 5.83 (0.000) 5.90 (0.127) 5.79 (0.000) 5.90 (0.127) 5.79 (0.000) 5.90 (0.127)
Exc. 3
6.62 (0.460) 6.24 (0.000) 6.51 (0.462) 6.22 (0.000) 6.51 (0.472) 6.22 (0.000) 6.51 (0.472)
Exc. 4
7.10 (0.273) 6.59 (0.000) 6.97 (0.231) 6.28 (0.004) 6.97 (0.231) 6.28 (0.004) 6.97 (0.231)
Exc. 5
8.35 (0.067) 7.28 (0.000) 7.27 (0.008) 6.56 (0.000) 7.27 (0.008) 6.56 (0.000) 7.27 (0.008)
Exc. 6
8.44 (0.129) 7.53 (0.006) 7.30 (0.022) 6.67 (0.006) 7.30 (0.022) 6.67 (0.006) 7.30 (0.022)
Exc. 7
8.79 (0.029) 8.21 (0.004) 7.77 (0.032) 6.97 (0.008) 7.77 (0.032) 6.97 (0.008) 7.77 (0.032)
Exc. 8
8.98 (0.067) 8.38 (0.001) 7.93 (0.009) 7.20 (0.000) 7.93 (0.009) 7.20 (0.000) 7.93 (0.009)
Exc. 9
9.15 (0.002) 8.68 (0.003) 8.07 (0.001) 7.66 (0.002) 8.07 (0.001) 7.66 (0.002) 8.07 (0.001)
Exc. 10 9.48 (0.061) 8.73 (0.000) 8.12 (0.029) 7.69 (0.002) 8.12 (0.029) 7.69 (0.002) 8.12 (0.029)
Exc. 11 9.59 (0.043) 9.03 (0.003) 8.19 (0.006) 7.83 (0.001) 8.19 (0.006) 7.83 (0.001) 8.19 (0.006)
Exc. 12 9.70 (0.020) 9.20 (0.005) 8.30 (0.061) 7.97 (0.002) 8.30 (0.060) 7.97 (0.002) 8.30 (0.060)
Exc. 13 9.91 (0.009) 9.55 (0.001) 8.34 (0.032) 8.04 (0.002) 8.35 (0.033) 8.05 (0.002) 8.35 (0.033)
Exc. 14 10.07 (0.018) 9.66 (0.008) 8.37 (0.031) 8.15 (0.000) 8.37 (0.032) 8.15 (0.000) 8.37 (0.032)
Exc. 15 10.16 (0.015) 9.67 (0.004) 8.41 (0.046) 8.24 (0.003) 8.42 (0.046) 8.19 (0.001) 8.42 (0.046)
Exp.
A00
7
TABLE XV. Table B3: The first 15 excited states for thymine using the CAS-srPBEGWS method
without and with added Rydberg basis functions and IP. Excitation energies in eV and oscillator
strengths in parentheses. Experimental data in last row.
TZVP
TZVP+Ryd-spd=2.5-5
A00
A0
A00
TZVP+Ryd-spd=2.5-4 TZVP+Ryd-spd=2.5-4+IP
Sym.
A0
A0
A00
A0
Exc. 1
5.29 (0.179)
5.10 (0.000) 5.34 (0.195) 4.96 (0.000) 5.34 (0.195) 4.96 (0.000) 5.34 (0.195)
Exc. 2
6.63 (0.069)
6.37 (0.000) 6.59 (0.076) 6.40 (0.001) 6.59 (0.077) 6.40 (0.001) 6.59 (0.077)
Exc. 3
6.71 (0.245)
7.19 (0.000) 6.62 (0.195) 6.42 (0.000) 6.62 (0.195) 6.42 (0.000) 6.62 (0.195)
Exc. 4
7.80 (0.409)
7.31 (0.000) 7.40 (0.079) 7.19 (0.000) 7.40 (0.079) 7.19 (0.000) 7.40 (0.079)
Exc. 5
8.53 (0.097)
7.47 (0.000) 7.57 (0.264) 7.28 (0.000) 7.57 (0.265) 7.28 (0.000) 7.59 (0.265)
Exc. 6
8.86 (0.033)
8.29 (0.000) 7.73 (0.121) 7.38 (0.000) 7.73 (0.120) 7.38 (0.000) 7.73 (0.120)
Exc. 7
9.12 (0.064)
8.52 (0.000) 8.04 (0.044) 7.45 (0.001) 8.04 (0.044) 7.45 (0.001) 8.04 (0.044)
Exc. 8
9.18 (0.070)
8.92 (0.001) 8.29 (0.009) 7.96 (0.000) 8.29 (0.009) 7.97 (0.000) 8.29 (0.009)
Exc. 9
9.39 (0.046)
8.97 (0.002) 8.34 (0.004) 8.16 (0.004) 8.34 (0.004) 8.16 (0.003) 8.34 (0.004)
Exc. 10 9.63 (0.122)
9.10 (0.001) 8.46 (0.008) 8.19 (0.003) 8.46 (0.008) 8.19 (0.003) 8.46 (0.008)
Exc. 11 9.76 (0.012)
9.50 (0.003) 8.56 (0.018) 8.29 (0.004) 8.56 (0.018) 8.29 (0.004) 8.56 (0.018)
Exc. 12 9.90 (0.004)
9.68 (0.026) 8.63 (0.012) 8.52 (0.001) 8.66 (0.013) 8.53 (0.001) 8.63 (0.012)
Exc. 13 9.99 (0.036)
9.87 (0.001) 8.72 (0.082) 8.54 (0.001) 8.72 (0.082) 8.54 (0.001) 8.72 (0.082)
Exc. 14 10.22 (0.098) 10.08 (0.010) 8.74 (0.059) 8.55 (0.000) 8.75 (0.060) 8.55 (0.000) 8.75 (0.060)
Exc. 15 10.25 (0.006) 10.12 (0.004) 8.85 (0.004) 8.56 (0.000) 8.85 (0.004) 8.56 (0.000) 8.85 (0.003)
Exp.
A00
8
TABLE XVI. Table B4: The first 15 excited states for uracil using the CAS-srPBEGWS method
without and with added Rydberg basis functions and IP. Excitation energies in eV and oscillator
strengths in parentheses. Experimental data in last row.
TZVP
TZVP+Ryd-spd=2.5-5
A00
A0
A00
TZVP+Ryd-spd=2.5-4 TZVP+Ryd-spd=2.5-4+IP
Sym
A0
A0
A00
A0
Exc. 1
5.52 (0.199)
4.93 (0.000) 5.51 (0.168) 4.92 (0.000) 5.51 (0.200) 4.92 (0.000) 5.51 (0.200)
Exc. 2
6.60 (0.056)
6.38 (0.000) 6.57 (0.057) 6.34 (0.000) 6.57 (0.057) 6.34 (0.000) 6.57 (0.057)
Exc. 3
6.83 (0.146)
7.20 (0.000) 6.75 (0.148) 6.64 (0.004) 6.75 (0.147) 6.64 (0.004) 6.75 (0.147)
Exc. 4
7.63 (0.466)
7.46 (0.001) 7.52 (0.386) 7.17 (0.000) 7.52 (0.386) 7.17 (0.000) 7.52 (0.386)
Exc. 5
8.80 (0.135)
7.50 (0.000) 7.72 (0.057) 7.46 (0.000) 7.72 (0.056) 7.46 (0.000) 7.72 (0.056)
Exc. 6
8.94 (0.114)
8.59 (0.000) 7.80 (0.072) 7.59 (0.009) 7.80 (0.073) 7.59 (0.009) 7.80 (0.073)
Exc. 7
9.01 (0.011)
8.94 (0.002) 8.09 (0.042) 7.63 (0.005) 8.09 (0.042) 7.63 (0.005) 8.09 (0.042)
Exc. 8
9.34 (0.093)
9.02 (0.016) 8.43 (0.003) 8.02 (0.001) 8.43 (0.004) 8.02 (0.001) 8.43 (0.004)
Exc. 9
9.76 (0.007)
9.13 (0.005) 8.70 (0.003) 8.47 (0.005) 8.70 (0.003) 8.47 (0.004) 8.70 (0.003)
Exc. 10 9.79 (0.026)
9.35 (0.001) 8.73 (0.024) 8.54 (0.000) 8.73 (0.023) 8.54 (0.000) 8.73 (0.023)
Exc. 11 9.96 (0.029)
9.76 (0.024) 8.77 (0.040) 8.58 (0.000) 8.77 (0.040) 8.58 (0.000) 8.77 (0.040)
Exc. 12 10.06 (0.023) 9.87 (0.008) 8.83 (0.161) 8.59 (0.000) 8.83 (0.161) 8.59 (0.000) 8.83 (0.161)
Exc. 13 10.19 (0.068) 9.95 (0.000) 8.90 (0.002) 8.75 (0.000) 8.90 (0.002) 8.75 (0.000) 8.90 (0.002)
Exc. 14 10.29 (0.004) 10.00 (0.001) 8.97 (0.011) 8.84 (0.011) 8.97 (0.010) 8.84 (0.011) 8.97 (0.010)
Exc. 15 10.61 (0.009) 10.07 (0.000) 8.99 (0.001) 8.95 (0.000) 8.99 (0.002) 8.95 (0.000) 8.99 (0.002)
Exp.
A00
9
TABLE XVII. Table C1: The first 25 excited states for acetone using the CAS-srPBEGWS metod
with two different values for maximum `: “sp” and “spd”. Excitation energies in eV and oscillator
strengths in parentheses.
TZVP + Ryd-sp=2.5-5
TZVP + Ryd-spd=2.5-5
Sym.
A1
B1
B2
A2
A1
B1
B2
Exc. 1
7.97 (0.002)
7.02 (0.034)
8.90 (0.000)
4.30
7.97 (0.006)
7.00 (0.030)
8.63 (0.007)
Exc. 2
8.95 (0.058)
7.93 (0.000)
9.61 (0.016)
7.84
8.57 (0.033)
7.93 (0.001)
8.90 (0.000)
Exc. 3
9.17 (0.198)
8.61 (0.000) 10.61 (0.052)
8.66
8.95 (0.070)
8.37 (0.016)
9.26 (0.003)
Exc. 4
9.42 (0.013)
8.98 (0.000) 11.05 (0.022)
8.94
9.16 (0.186)
8.61 (0.004)
9.55 (0.002)
Exc. 5
9.64 (0.002)
9.22 (0.001) 11.23 (0.006)
9.38
9.23 (0.023)
8.73 (0.011)
9.58 (0.013)
Exc. 6
9.96 (0.001)
9.40 (0.000) 11.27 (0.006)
9.62
9.43 (0.014)
8.99 (0.000)
9.85 (0.002)
Exc. 7 10.51 (0.020) 9.51 (0.001) 11.66 (0.038)
9.81
9.54 (0.010)
9.13 (0.004) 10.45 (0.002)
Exc. 8 10.55 (0.044) 9.63 (0.000) 11.87 (0.015)
9.92
9.66 (0.004)
9.25 (0.002) 10.56 (0.057)
Exc. 9 10.80 (0.138) 9.67 (0.004) 11.91 (0.027)
10.41
9.82 (0.010)
9.29 (0.004) 10.97 (0.004)
Exc. 10 11.05 (0.097) 9.81 (0.006) 11.98 (0.054)
10.51
9.98 (0.010)
9.41 (0.000) 11.03 (0.020)
Exc. 11 11.46 (0.003) 9.95 (0.000) 12.10 (0.001)
10.66
10.39 (0.023) 9.48 (0.002) 11.07 (0.000)
Exc. 12 11.63 (0.000) 9.99 (0.005) 12.20 (0.009)
10.98
10.48 (0.027) 9.55 (0.001) 11.28 (0.006)
Exc. 13 11.96 (0.001) 10.27 (0.005) 12.33 (0.005)
11.31
10.74 (0.110) 9.56 (0.023) 11.38 (0.008)
Exc. 14 12.02 (0.001) 10.31 (0.052) 12.36 (0.007)
11.49
10.75 (0.004) 9.64 (0.000) 11.64 (0.046)
Exc. 15 12.07 (0.001) 10.65 (0.152) 12.47 (0.004)
11.69
11.27 (0.000) 9.70 (0.000) 11.73 (0.005)
Exc. 16 12.31 (0.002) 10.89 (0.087) 12.58 (0.004)
12.09
11.34 (0.013) 9.76 (0.001) 11.78 (0.003)
Exc. 17 12.38 (0.016) 11.70 (0.020) 12.65 (0.014)
12.32
11.61 (0.027) 9.84 (0.003) 11.89 (0.029)
Exc. 18 12.60 (0.024) 11.83 (0.001) 12.94 (0.072)
12.48
11.63 (0.000) 9.94 (0.001) 11.95 (0.007)
Exc. 19 12.61 (0.020) 11.97 (0.043) 12.99 (0.000)
12.54
11.84 (0.007) 9.96 (0.000) 12.00 (0.008)
Exc. 20 12.69 (0.053) 12.08 (0.003) 13.05 (0.021)
12.65
11.91 (0.000) 10.13 (0.000) 12.03 (0.027)
Exc. 21 12.83 (0.006) 12.22 (0.003) 13.18 (0.010)
12.74
11.93 (0.002) 10.25 (0.004) 12.10 (0.002)
Exc. 22 12.98 (0.005) 12.60 (0.003) 13.31 (0.028)
12.93
12.08 (0.001) 10.31 (0.056) 12.16 (0.001)
Exc. 23 13.10 (0.010) 12.68 (0.061) 13.37 (0.000)
13.06
12.13 (0.001) 10.43 (0.033) 12.19 (0.001)
Exc. 24 13.18 (0.021) 12.76 (0.105) 13.44 (0.005)
13.16
12.19 (0.036) 10.61 (0.123) 12.23 (0.001)
Exc. 25 13.21 (0.007) 12.86 (0.056) 13.50 (0.032)
13.30
12.22 (0.000) 10.82 (0.006) 12.25 (0.000)
Exp.
7.41, 7.80
6.36, 7.49,
8.09
8.17
4.43, 7.36
7.41, 7.80
6.36, 7.49,
8.09
8.17
A2
4.43, 7.36
10
TABLE XVIII. Table C2: The first 25 excited states for acetone using CAS-srPBEGWS with two
different values maximum `: “spd” and “spdf ”. Excitation energies in eV and oscillator strengths
in parentheses.
TZVP + Ryd-spd=2.5-5
TZVP + Ryd-spdf=2.5-5
Sym.
A1
B1
B2
A2
Exc. 1
7.97 (0.006)
7.00 (0.030)
8.63 (0.007)
4.29
7.97 (0.006) 7.00 (0.029) 8.63 (0.007)
Exc. 2
8.57 (0.033)
7.93 (0.001)
8.90 (0.000)
7.84
8.57 (0.038) 7.93 (0.001) 8.89 (0.000)
Exc. 3
8.95 (0.070)
8.37 (0.016)
9.26 (0.003)
8.56
8.94 (0.085) 8.37 (0.016) 9.19 (0.000)
Exc. 4
9.16 (0.186)
8.61 (0.004)
9.55 (0.002)
8.66
9.13 (0.145) 8.60 (0.003) 9.26 (0.003)
Exc. 5
9.23 (0.023)
8.73 (0.011)
9.58 (0.013)
8.95
9.21 (0.007) 8.72 (0.009) 9.51 (0.000)
Exc. 6
9.43 (0.014)
8.99 (0.000)
9.85 (0.002)
9.22
9.22 (0.022) 8.99 (0.001) 9.55 (0.003)
Exc. 7
9.54 (0.010)
9.13 (0.004) 10.45 (0.002)
9.39
9.23 (0.021) 9.12 (0.001) 9.58 (0.013)
Exc. 8
9.66 (0.004)
9.25 (0.002) 10.56 (0.057)
9.53
9.43 (0.011) 9.16 (0.005) 9.78 (0.000)
Exc. 9
9.82 (0.010)
9.29 (0.004) 10.97 (0.004)
9.63
9.52 (0.001) 9.22 (0.000) 9.86 (0.002)
Exc. 10 9.98 (0.010)
9.41 (0.000) 11.03 (0.020)
9.80
9.53 (0.002) 9.25 (0.001) 10.22 (0.000)
Exc. 11 10.39 (0.023) 9.48 (0.002) 11.07 (0.000)
9.81
9.54 (0.009) 9.29 (0.004) 10.47 (0.005)
Exc. 12 10.48 (0.027) 9.55 (0.001) 11.28 (0.006)
9.93
9.66 (0.004) 9.41 (0.000) 10.55 (0.055)
Exc. 13 10.74 (0.110) 9.56 (0.023) 11.38 (0.008)
10.34
9.80 (0.002) 9.46 (0.000) 10.94 (0.003)
Exc. 14 10.75 (0.004) 9.64 (0.000) 11.64 (0.046)
10.38
9.82 (0.000) 9.49 (0.004) 10.98 (0.001)
Exc. 15 11.27 (0.000) 9.70 (0.000) 11.73 (0.005)
10.62
9.82 (0.011) 9.53 (0.000) 11.02 (0.017)
Exc. 16 11.34 (0.013) 9.76 (0.001) 11.78 (0.003)
10.91
9.98 (0.010) 9.55 (0.001) 11.24 (0.008)
Exc. 17 11.61 (0.027) 9.84 (0.003) 11.89 (0.029)
11.23
10.29 (0.001) 9.57 (0.003) 11.37 (0.007)
Exc. 18 11.63 (0.000) 9.94 (0.001) 11.95 (0.007)
11.27
10.34 (0.003) 9.64 (0.000) 11.63 (0.052)
Exc. 19 11.84 (0.007) 9.96 (0.000) 12.00 (0.008)
11.37
10.39 (0.021) 9.70 (0.000) 11.69 (0.000)
Exc. 20 11.91 (0.000) 10.13 (0.000) 12.03 (0.027)
11.44
10.48 (0.025) 9.70 (0.001) 11.75 (0.003)
Exc. 21 11.93 (0.002) 10.25 (0.004) 12.10 (0.002)
11.69
10.73 (0.108) 9.79 (0.006) 11.79 (0.000)
Exc. 22 12.08 (0.001) 10.31 (0.056) 12.16 (0.001)
11.90
10.77 (0.004) 9.82 (0.001) 11.81 (0.005)
Exc. 23 12.13 (0.001) 10.43 (0.033) 12.19 (0.001)
12.10
11.21 (0.000) 9.85 (0.001) 11.90 (0.011)
Exc. 24 12.19 (0.036) 10.61 (0.123) 12.23 (0.001)
12.16
11.24 (0.000) 9.95 (0.001) 11.91 (0.010)
Exc. 25 12.22 (0.000) 10.82 (0.006) 12.25 (0.000)
12.21
11.31 (0.013) 9.96 (0.000) 11.94 (0.010)
Exp.
7.41, 7.80
6.36, 7.49,
8.09
8.17
4.43, 7.36
A1
7.41, 7.80
B1
6.36, 7.49,
8.09
B2
8.17
A2
4.43, 7.36