7.38. Core-level spectroscopy with coupled cluster methods¶
The equation of motion coupled cluster method and its similarity transformed version provides an easy way to directly calculate core-ionization and core-excitation energies. Currently, the core-level spectroscopy with EOM-CCSD and STEOM-CCSD is only available for closed shell systems.
7.38.1. Core-ionization¶
One can obtain core-ionized states if one calculates a large no of roots. The ORCA implementation of IP-EOM-CCSD, however, allows one to directly target the ionization from the core-orbitals. A typical IP-EOM-CCSD input file for the acetic acid will look like
!IP-EOM-CCSD ExtremeSCF cc-pvtz
!NoFrozencore
%maxcore 5000
%mdci
nroots 4 #no of roots
CVSORB 0,3 #orbital considered for core-valence separation
FollowCIS true # Follow the initial guess orbital
CVSEP true # Core valence separation
DoCVS true # Core valence separation (currently both the option needs to be true)
DoCore true # Directly target the core
corehole 0 # The state from which it will count the roots
printlevel 3 # the printing options
maxiter 500 # no of iteration, generally requires larger no of roots
end
*xyz 0 1
C -6.7624010562 0.1328615492 0.0389382700
C -5.3564667033 0.2819965475 -0.5188248498
H -6.9983743824 1.0019615710 0.6510029634
H -7.4924880320 0.0542210905 -0.7741766747
H -6.8380664832 -0.7720291637 0.6519904379
O -4.9303467983 -0.7518088469 -1.3223158759
H -5.6257914271 -1.4265892921 -1.4015111180
O -4.6208051175 1.2132365445 -0.3081931529
*
The output of it will be
----------------------
EOM-CCSD RESULTS (RHS)
----------------------
IROOT= 1: 19.902202 au 541.566 eV 4368028.3 cm**-1
Amplitude Excitation
-0.673297 0 -> x
Percentage singles character= 82.93
IROOT= 2: 19.842487 au 539.942 eV 4354922.4 cm**-1
Amplitude Excitation
0.672818 1 -> x
Percentage singles character= 82.71
IROOT= 3: 10.891843 au 296.382 eV 2390483.3 cm**-1
Amplitude Excitation
-0.669218 2 -> x
Percentage singles character= 81.11
IROOT= 4: 10.754926 au 292.656 eV 2360433.5 cm**-1
Amplitude Excitation
-0.670254 3 -> x
Percentage singles character= 81.57
The option ‘DoCore true’ starts the counting of the roots from ‘corehole’ upwards. The default is ‘DoCore false’ and it counts the root from the HOMO downwards. The ‘corehole 0’ starts the counting from the first occupied orbital which is the oxygen K-edge in this case. One can directly target the carbon K-edge in this case by putting ‘corehole 2’.
!IP-EOM-CCSD ExtremeSCF cc-pvtz
!NoFrozencore
%maxcore 5000
%mdci
nroots 2 #no of roots
CVSORB 2,3 #orbital considered for core-valence separation
FollowCIS true # Follow the initial guess orbital
CVSEP true # Core valence separation
DoCVS true # Core valence separation (currently both the option needs to be true)
DoCore true # Directly target the core
corehole 2 # The state from which it will count the roots
printlevel 3 # the printing options
maxiter 500 # no of iteration, generally requires larger no of roots
end
*xyz 0 1
C -6.7624010562 0.1328615492 0.0389382700
C -5.3564667033 0.2819965475 -0.5188248498
H -6.9983743824 1.0019615710 0.6510029634
H -7.4924880320 0.0542210905 -0.7741766747
H -6.8380664832 -0.7720291637 0.6519904379
O -4.9303467983 -0.7518088469 -1.3223158759
H -5.6257914271 -1.4265892921 -1.4015111180
O -4.6208051175 1.2132365445 -0.3081931529
*
The output of it will be
----------------------
EOM-CCSD RESULTS (RHS)
----------------------
IROOT= 1: 10.891843 au 296.382 eV 2390483.3 cm**-1
Amplitude Excitation
0.669218 2 -> x
Percentage singles character= 81.11
IROOT= 2: 10.754926 au 292.656 eV 2360433.5 cm**-1
Amplitude Excitation
-0.670254 3 -> x
Percentage singles character= 81.57
Now, the core-ionized states remains embedded in the high density of doubly ionized valence states that form the continuum. This leads to severe convergence problems. One easy way to overcome this is to use the core-valence separation approximation which is turned on by the two keywords ‘CVSEP true’ and ‘DoCVS true’. The orbitals from which the contributions are not neglected for the core-valence separation are set by ‘CVSORB initial,final’. It is generally a good idea to include all the core orbitals corresponding to a particular element if one is interested in the ionization from any of the core orbitals for the particular element. In the second example both the carbon core-orbitals are included in the ‘CVSORB 2,3’. A ‘bt-PNO-IP-EOM-CCSD’ input file for the same example will look like
!bt-PNO-IP-EOM-CCSD ExtremeSCF cc-pvtz cc-pvtz/c
!NoFrozencore
%maxcore 5000
%mdci
nroots 4
CVSORB 0,3
FollowCIS true
CVSEP true
DoCVS true
DoCore true
DoRECAN true # recanonilize the occupied space before the EOM step
corehole 0
printlevel 3
maxiter 500
end
*xyz 0 1
C -6.7624010562 0.1328615492 0.0389382700
C -5.3564667033 0.2819965475 -0.5188248498
H -6.9983743824 1.0019615710 0.6510029634
H -7.4924880320 0.0542210905 -0.7741766747
H -6.8380664832 -0.7720291637 0.6519904379
O -4.9303467983 -0.7518088469 -1.3223158759
H -5.6257914271 -1.4265892921 -1.4015111180
O -4.6208051175 1.2132365445 -0.3081931529
*
The output of it will be
IROOT= 1: 19.901845 au 541.557 eV 4367950.0 cm**-1
Amplitude Excitation
-0.673298 0 -> x
Percentage singles character= 82.93
IROOT= 2: 19.842152 au 539.932 eV 4354848.9 cm**-1
Amplitude Excitation
0.672832 1 -> x
Percentage singles character= 82.72
IROOT= 3: 10.892369 au 296.396 eV 2390598.7 cm**-1
Amplitude Excitation
-0.669213 2 -> x
Percentage singles character= 81.11
IROOT= 4: 10.754951 au 292.657 eV 2360438.9 cm**-1
Amplitude Excitation
-0.670244 3 -> x
Percentage singles character= 81.56
The results are in excellent agreement with the canonical one. A DLPNO variant for the same example will look like
!IP-EOM-DLPNO-CCSD ExtremeSCF cc-pvtz cc-pvtz/c autoaux def2/J TightPNO pal16
!NoFrozencore
%maxcore 5000
%mdci
nroots 4
CVSORB 0,3
FollowCIS true
CVSEP true
DoCVS true
DoCore true
corehole 0
printlevel 3
maxiter 500
end
*xyz 0 1
C -6.7624010562 0.1328615492 0.0389382700
C -5.3564667033 0.2819965475 -0.5188248498
H -6.9983743824 1.0019615710 0.6510029634
H -7.4924880320 0.0542210905 -0.7741766747
H -6.8380664832 -0.7720291637 0.6519904379
O -4.9303467983 -0.7518088469 -1.3223158759
H -5.6257914271 -1.4265892921 -1.4015111180
O -4.6208051175 1.2132365445 -0.3081931529
*
The output of it will be
----------------------
EOM-CCSD RESULTS (RHS)
----------------------
IROOT= 1: 19.945319 au 542.740 eV 4377491.5 cm**-1
Amplitude Excitation
0.678788 0 -> x
Percentage singles character= 101.06
IROOT= 2: 19.890529 au 541.249 eV 4365466.5 cm**-1
Amplitude Excitation
0.679716 1 -> x
Percentage singles character= 101.02
IROOT= 3: 10.912292 au 296.939 eV 2394971.3 cm**-1
Amplitude Excitation
0.672646 2 -> x
Percentage singles character= 101.22
IROOT= 4: 10.792478 au 293.678 eV 2368675.1 cm**-1
Amplitude Excitation
0.674795 3 -> x
Percentage singles character= 101.14
Although the error in the absolute IP values are as large as 1 eV, the so-called ‘chemical shift’ i.e. the difference between the IP value of two different atoms of the same elements are reasonably correct.
7.38.2. Core-Excitation¶
The STEOM-CCSD approach provides an efficient and accurate way to do the K-edge core-excitation spectroscopy. A typical input file for the STEOM-CCSD will look like
!STEOM-CCSD ExtremeSCF aug-cc-pCVQZ Bohrs NoFrozencore
%mdci
nroots 10
CVSORB 6,6 # should always be HOMO
FollowCIS true
CVSEP true
DoCVS true
DoCore true
DoSimpleDens False # use exact STEOM transition moment
corehole 0
maxiter 500
NDAV 80
printlevel 3
end
*xyz 0 1
O 0 0 0.913973
C 0 0 -1.218243
*
The output will be
------------------
STEOM-CCSD RESULTS
------------------
IROOT= 1: 19.686174 au 535.688 eV 4320615.8 cm**-1
Amplitude Excitation
-0.215317 6 -> 7
-0.211332 6 -> 8
-0.442546 6 -> 11
0.330607 6 -> 12
0.568497 6 -> 15
-0.506486 6 -> 16
Ground state amplitude: 0.000000
Percentage Active Character 97.59
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
-0.177019 0 -> 8
0.346366 0 -> 9
0.590044 0 -> 11
0.464534 0 -> 12
-0.214399 0 -> 14
0.338336 0 -> 15
-0.146649 0 -> 20
0.169124 0 -> 21
-0.192930 0 -> 24
IROOT= 2: 19.686174 au 535.688 eV 4320615.8 cm**-1
Amplitude Excitation
0.211332 6 -> 7
-0.215317 6 -> 8
0.330607 6 -> 11
0.442546 6 -> 12
-0.506486 6 -> 15
-0.568497 6 -> 16
Ground state amplitude: 0.000000
Percentage Active Character 97.59
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
0.346366 0 -> 8
0.177019 0 -> 9
0.464534 0 -> 11
-0.590044 0 -> 12
0.338336 0 -> 14
0.214399 0 -> 15
0.169124 0 -> 20
0.146649 0 -> 21
-0.192930 0 -> 23
IROOT= 3: 19.865373 au 540.564 eV 4359945.5 cm**-1
Amplitude Excitation
-0.571289 6 -> 9
0.792679 6 -> 10
0.137627 6 -> 13
-0.112257 6 -> 17
Ground state amplitude: -0.000591
Percentage Active Character 97.37
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
-0.900242 0 -> 7
-0.116863 0 -> 13
-0.375672 0 -> 18
0.128317 0 -> 19
IROOT= 4: 19.909335 au 541.761 eV 4369594.0 cm**-1
Amplitude Excitation
0.340300 6 -> 7
0.704671 6 -> 8
-0.338179 6 -> 11
0.511324 6 -> 12
Ground state amplitude: 0.000000
Percentage Active Character 99.71
Amplitude Excitation in Canonical Basis
0.101796 0 -> 8
-0.793309 0 -> 9
0.482364 0 -> 11
0.160972 0 -> 12
-0.209491 0 -> 15
-0.128207 0 -> 23
-0.188543 0 -> 24
IROOT= 5: 19.909335 au 541.761 eV 4369594.0 cm**-1
Amplitude Excitation
-0.704671 6 -> 7
0.340300 6 -> 8
0.511324 6 -> 11
0.338179 6 -> 12
Ground state amplitude: 0.000000
Percentage Active Character 99.71
Amplitude Excitation in Canonical Basis
-0.793309 0 -> 8
-0.101796 0 -> 9
0.160972 0 -> 11
-0.482364 0 -> 12
-0.209491 0 -> 14
-0.188543 0 -> 23
0.128207 0 -> 24
IROOT= 6: 19.914772 au 541.909 eV 4370787.3 cm**-1
Amplitude Excitation
-0.804799 6 -> 9
-0.557108 6 -> 10
-0.125228 6 -> 13
0.119745 6 -> 17
Ground state amplitude: 0.000364
Percentage Active Character 97.38
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
0.934227 0 -> 10
-0.273222 0 -> 13
0.144269 0 -> 18
-0.159846 0 -> 22
IROOT= 7: 19.966983 au 543.329 eV 4382246.2 cm**-1
Amplitude Excitation
0.190413 6 -> 10
-0.954987 6 -> 13
0.113662 6 -> 22
Ground state amplitude: 0.000138
Percentage Active Character 94.88
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
-0.190758 0 -> 7
0.246282 0 -> 10
0.890287 0 -> 13
0.198835 0 -> 18
0.170782 0 -> 19
0.180954 0 -> 25
IROOT= 8: 19.981194 au 543.716 eV 4385365.2 cm**-1
Amplitude Excitation
0.513702 6 -> 7
0.587528 6 -> 11
0.608754 6 -> 15
Ground state amplitude: -0.000000
Percentage Active Character 98.92
Amplitude Excitation in Canonical Basis
0.391272 0 -> 8
-0.123811 0 -> 9
-0.308174 0 -> 12
-0.796992 0 -> 14
0.173278 0 -> 15
-0.202678 0 -> 20
0.130010 0 -> 29
IROOT= 9: 19.981194 au 543.716 eV 4385365.2 cm**-1
Amplitude Excitation
-0.513700 6 -> 8
0.587528 6 -> 12
0.608756 6 -> 16
Ground state amplitude: -0.000000
Percentage Active Character 98.92
Amplitude Excitation in Canonical Basis
0.123796 0 -> 8
0.391276 0 -> 9
0.308177 0 -> 11
-0.173290 0 -> 14
-0.796989 0 -> 15
-0.202680 0 -> 21
0.130002 0 -> 28
IROOT= 10: 20.005026 au 544.364 eV 4390595.8 cm**-1
Amplitude Excitation
0.997209 6 -> 14
Ground state amplitude: 0.000000
Percentage Active Character 99.44
Amplitude Excitation in Canonical Basis
0.980462 0 -> 16
-0.142106 0 -> 26
Excitation from a particular core orbital can be considered currently. In the present case it is the 1S orbital of oxygen. The required orbital can be specified using the keyword ‘corehole’. For the oxygen 1S it should be ‘corehole 0’. The carbon 1S can be specified with ‘corehole 1’
!STEOM-CCSD ExtremeSCF aug-cc-pCVQZ Bohrs NoFrozencore
%mdci
nroots 10
CVSORB 6,6 # should always be HOMO
FollowCIS true
CVSEP true
DoCVS true
DoCore true
DoSimpleDens False # use exact STEOM transition moment
corehole 1
maxiter 500
NDAV 80
printlevel 3
end
*xyz 0 1
O 0 0 0.913973
C 0 0 -1.218243
*
It will give the carbon K-edge spectra as follows
------------------
STEOM-CCSD RESULTS
------------------
IROOT= 1: 10.569902 au 287.622 eV 2319825.3 cm**-1
Amplitude Excitation
0.429677 6 -> 8
0.132429 6 -> 11
0.880021 6 -> 16
Ground state amplitude: -0.000000
Percentage Active Character 98.73
Amplitude Excitation in Canonical Basis
0.158027 1 -> 8
0.273564 1 -> 9
-0.354592 1 -> 11
-0.668565 1 -> 12
0.184004 1 -> 14
0.308132 1 -> 15
0.112105 1 -> 20
0.195715 1 -> 21
0.134711 1 -> 23
0.275743 1 -> 24
-0.140527 1 -> 29
IROOT= 2: 10.569902 au 287.622 eV 2319825.3 cm**-1
Amplitude Excitation
0.429686 6 -> 7
-0.132345 6 -> 12
0.880029 6 -> 15
Ground state amplitude: -0.000000
Percentage Active Character 98.73
Amplitude Excitation in Canonical Basis
-0.273564 1 -> 8
0.158027 1 -> 9
0.668565 1 -> 11
-0.354592 1 -> 12
-0.308132 1 -> 14
0.184004 1 -> 15
-0.195715 1 -> 20
0.112105 1 -> 21
-0.275744 1 -> 23
0.134711 1 -> 24
0.140527 1 -> 28
IROOT= 3: 10.807563 au 294.089 eV 2371985.9 cm**-1
Amplitude Excitation
-0.759965 6 -> 9
-0.366007 6 -> 10
0.514219 6 -> 13
Ground state amplitude: 0.000746
Percentage Active Character 97.59
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
0.865305 1 -> 7
-0.246321 1 -> 10
-0.230712 1 -> 13
0.312193 1 -> 18
0.113059 1 -> 25
IROOT= 4: 10.840510 au 294.985 eV 2379217.0 cm**-1
Amplitude Excitation
-0.813493 6 -> 7
0.345752 6 -> 12
0.449353 6 -> 15
Ground state amplitude: 0.000000
Percentage Active Character 98.33
Amplitude Excitation in Canonical Basis
0.803886 1 -> 8
-0.349815 1 -> 9
0.318183 1 -> 11
-0.124975 1 -> 12
0.186068 1 -> 14
-0.239028 1 -> 23
IROOT= 5: 10.840510 au 294.985 eV 2379217.0 cm**-1
Amplitude Excitation
0.813492 6 -> 8
0.345726 6 -> 11
-0.449377 6 -> 16
Ground state amplitude: 0.000000
Percentage Active Character 98.33
Amplitude Excitation in Canonical Basis
0.349809 1 -> 8
0.803873 1 -> 9
0.124977 1 -> 11
0.318188 1 -> 12
0.186103 1 -> 15
-0.239038 1 -> 24
IROOT= 6: 10.845730 au 295.127 eV 2380362.5 cm**-1
Amplitude Excitation
0.116483 2 -> 136
0.438618 6 -> 9
-0.879925 6 -> 10
0.106779 6 -> 19
Ground state amplitude: 0.000706
Percentage Active Character 96.67
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
-0.244947 1 -> 7
-0.900149 1 -> 10
-0.170451 1 -> 13
-0.205137 1 -> 18
-0.192328 1 -> 22
0.116486 3 -> 136
IROOT= 7: 10.906409 au 296.778 eV 2393680.2 cm**-1
Amplitude Excitation
0.113733 2 -> 136
-0.397156 6 -> 9
-0.234927 6 -> 10
-0.815420 6 -> 13
-0.293892 6 -> 17
0.126290 6 -> 30
Ground state amplitude: -0.000417
Percentage Active Character 87.78
Warning:: the state may have not converged with respect to active space
-------------------- Handle with Care --------------------
Amplitude Excitation in Canonical Basis
0.219961 1 -> 7
-0.178103 1 -> 10
0.883509 1 -> 13
-0.239553 1 -> 18
-0.188730 1 -> 19
-0.161616 1 -> 25
0.113733 3 -> 136
IROOT= 8: 10.926332 au 297.321 eV 2398052.6 cm**-1
Amplitude Excitation
-0.365429 6 -> 8
0.927126 6 -> 11
Ground state amplitude: 0.000000
Percentage Active Character 99.66
Amplitude Excitation in Canonical Basis
-0.129108 1 -> 8
-0.295033 1 -> 9
0.119506 1 -> 11
0.302428 1 -> 12
0.353687 1 -> 14
0.777349 1 -> 15
0.200807 1 -> 21
IROOT= 9: 10.926332 au 297.321 eV 2398052.8 cm**-1
Amplitude Excitation
0.365449 6 -> 7
0.927109 6 -> 12
Ground state amplitude: 0.000000
Percentage Active Character 99.66
Amplitude Excitation in Canonical Basis
-0.295006 1 -> 8
0.129096 1 -> 9
0.302490 1 -> 11
-0.119530 1 -> 12
0.777245 1 -> 14
-0.353640 1 -> 15
0.201146 1 -> 20
IROOT= 10: 10.944288 au 297.809 eV 2401993.5 cm**-1
Amplitude Excitation
0.998859 6 -> 14
Ground state amplitude: -0.000000
Percentage Active Character 99.77
Amplitude Excitation in Canonical Basis
-0.983120 1 -> 16
-0.104304 1 -> 33
The core-valence separation should be used similar to that in the core-ionization. The only difference is that the natural orbital based active space selection scheme in STEOM-CCSD always rotate the particular core orbital to the HOMO. Therefore, CVSORB should always be HOMO in STEOM-CCSD irrespective of the core-hole. One should use the exact STEOM-CCSD transition moment by using DoSimpleDens False. Fig. 7.40 presents the STEOM-CCSD oxygen K-edge spectra in thymine.
Fig. 7.39 Comparison of theoretical and experimental X-ray absorption spectra of oxygen K-edge in thymine. The simulated spectrum is shifted by -3.7 eV to align with the experimental spectrum.¶
One can interpret the results in terms of NTOs caculated from STEOM-CC eigen vectors
Fig. 7.40 Natural transition orbitals (ntos) for the oxygen K edge spectrum of thymine. All the core EE values mentioned are in eV and provided in the format (EE,Oscillator Strength).¶