Electronic structure of CmF
We want to investigate the structure of the diatomic molecule CmF (Curium Fluoride). Following [Mochizuki_JCP2003] we assume that curium has valence configuration in \(5f^77s^2\) in the molecule, in particular since curium is a late actinide and thus quite similar to lanthanides. The molecule will therefore have an atomic-like open \(5f^7\) shell. This may cause convergence problems since DIRAC by default assumes open-shell orbitals to have energies above close-shell ones, which is likely to not be the case here. In order to guide DIRAC we shall therefore employ the following strategy:
We calculate the curium atom.
We import the \(5f^7\) orbitals into the molecular calculation, place them in the open shell and keep them frozen in a first SCF calculation.
We then freeze all closed-shell orbitals, allowing the \(5f^7\) orbitals to relax in the molecule.
The curium atom
For ease of analysis we shall impose linear symmetry also in the atomic calculation. We achieve this through the introduction of a ghost center
2
Cm 0.00 0.00 0.00
Cm.GH 0.00 0.00 10.00
The ground state configuration of curium (Cm, Z=96) is \([Rn]5f^76d^17s^2\) (see here for general information and also here), so with two open shells. We specify this configuration by using the keyword .KPSELE:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**ANALYZE
.MULPOP
*MULPOP
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
88
.OPEN SHELL
2
7/14
1/10
.OPENFAC
1.0
.KPSELEC
7
-1 1 -2 2 -3 3 -4
14 10 20 12 18 6 8
0 0 0 0 0 6 8
0 0 0 4 6 0 0
**MOLECULE
*BASIS
.DEFAULT
dyall.cv3z
.SPECIAL
Cm.GH NOBASIS
*END OF
We run the calculation using:
pam --inp=Cm --mol=Cm.xyz --get "DFCOEF=cf.Cm"
This calculation converges smoothly in 13 iterations, and we are now ready to attack the molecule.
The molecular calculation:
1. Importing curium \(5f\) orbitals
The menu file for the first step of the molecular calculation reads
**DIRAC
.WAVE FUNCTION
.ANALYZE
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
98
.OPEN SHELL
1
7/14
.OPENFAC
1.0
.OWNBAS
.PROJEC
1
AFCMXX
1
45..51
.FROZEN
50..56
**ANALYZE
.MULPOP
*MULPOP
.LABEL
SHELL
**MOLECULE
*BASIS
.DEFAULT
dyall.cv3z
**END OF
In the *SCF section it may be noted that we specify 49 closed-shell orbitals, followed by seven open-shell ones, with seven active electrons. Next we specify that we will take orbitals 45 to 51 from the coefficient file of the cerium atom, that we name AFCMXX, put them in position 50 to 56 and keep them frozen. This also implies that we are effectively doing a closed-shell calculation, which helps convergence as well. We run the calculation using the command:
pam --mw=200 --inp=CmF_5f_frz --mol=CmF.xyz --put "cf.Cm=AFCMXX" --get "DFCOEF=cf.CmF_5f_frz"
The calculation converges in 18 iterations. Here is a snippet from the population analysis:
* Electronic eigenvalue no. 49: -0.2232533658894 (Occupation : f = 1.0000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 p L Cm 1 d L Cm 1 f L F 1 s L F 1 p
-----------------------------------------------------------------------------------------------------------------
alpha 0.9978 | 0.8852 0.0413 0.0709 0.0002 -0.0008 0.0007
beta 0.0022 | 0.0000 0.0021 0.0001 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 50: -0.5359098383804 (Occupation : f = 0.5000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.4286 | 0.4284 0.0003
beta 0.5714 | 0.5712 0.0002
* Electronic eigenvalue no. 51: -0.5359098383796 (Occupation : f = 0.5000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.7141 | 0.7140 0.0001
beta 0.2859 | 0.2856 0.0003
* Electronic eigenvalue no. 52: -0.5359098384723 (Occupation : f = 0.5000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.1432 | 0.1428 0.0004
beta 0.8568 | 0.8568 0.0000
* Electronic eigenvalue no. 53: -0.4840413469146 (Occupation : f = 0.5000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.2859 | 0.2856 0.0003
beta 0.7141 | 0.7140 0.0001
* Electronic eigenvalue no. 54: -0.4840413470277 (Occupation : f = 0.5000) m_j= -7/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.0004 | 0.0000 0.0004
beta 0.9996 | 0.9996 0.0000
* Electronic eigenvalue no. 55: -0.4840413468813 (Occupation : f = 0.5000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.5714 | 0.5712 0.0002
beta 0.4286 | 0.4284 0.0002
* Electronic eigenvalue no. 56: -0.4840413469351 (Occupation : f = 0.5000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.8569 | 0.8568 0.0001
beta 0.1431 | 0.1428 0.0003
It shows that the upper closed-shell orbital (49) is dominated by cerium \((7)s\). The open-shell cerium \(5f\) orbitals are frozen and keep their energies from the atomic calculation, with \(5f_{5/2}\) and \(5f_{7/2}\) orbitals having energies -0.5359 and -0.4840 \(E_h\), respectively.
2. Relaxing the curium \(5f\) orbitals
We now want to relax the curium \(5f\) orbitals in the molecule whilst keeping the other orbitals frozen. This step is not necessarily requred, but allows us to see how the \(5f\) orbital energies evolve. We use the following input file
**DIRAC
.WAVE FUNCTION
.ANALYZE
**WAVE FUNCTION
.SCF
*SCF
.MAXITR
80
.CLOSED SHELL
98
.OPEN SHELL
1
7/14
.OPENFAC
1.0
.PROJEC
1
DFCOEF
1
1..49
.FROZEN
1..49
**ANALYZE
.MULPOP
*MULPOP
.LABEL
SHELL
**MOLECULE
*BASIS
.DEFAULT
dyall.cv3z
**END OF
It will be seen that we now import the coefficient file of the previous run and freeze all closed-shell orbitals. We use the command:
pam --mw=200 --inp=CmF_5f_frz --mol=CmF.xyz --put "cf.CmF_5f_frz=DFCOEF" --get "DFCOEF=cf.CmF_5f_relax"
The calculation converges in 14 iterations. Again a snippet from the Mulliken population analysis:
* Electronic eigenvalue no. 46: -0.5957664135907 (Occupation : f = 1.0000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 p L Cm 1 d L Cm 1 f L Cm 1 g L Cm 1 h L F 1 s
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.4347 | 0.0063 0.0140 0.0279 0.0021 0.0006 0.0001 0.0023
beta 0.5653 | 0.0000 0.0089 0.0209 0.0061 0.0037 0.0008 0.0000
Gross | L F 1 p L F 1 d
--------------------------------------
alpha | 0.3812 0.0001
beta | 0.5244 0.0004
* Electronic eigenvalue no. 47: -0.5919466306963 (Occupation : f = 1.0000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 p L Cm 1 d L Cm 1 f L Cm 1 g L Cm 1 h L F 1 s
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.5583 | 0.0086 0.0284 0.0362 0.0027 0.0008 0.0002 0.0022
beta 0.4417 | 0.0000 0.0068 0.0151 0.0045 0.0029 0.0006 0.0000
Gross | L F 1 p L F 1 d
--------------------------------------
alpha | 0.4792 0.0001
beta | 0.4116 0.0002
* Electronic eigenvalue no. 48: -0.5915916419691 (Occupation : f = 1.0000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 p L Cm 1 d L Cm 1 f L Cm 1 g L Cm 1 h L F 1 p L F 1 d
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.0001 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
beta 0.9999 | 0.0088 0.0357 0.0105 0.0066 0.0014 0.9363 0.0005
* Electronic eigenvalue no. 49: -0.2232533658894 (Occupation : f = 1.0000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 p L Cm 1 d L Cm 1 f L F 1 s L F 1 p
-----------------------------------------------------------------------------------------------------------------
alpha 0.9978 | 0.8852 0.0413 0.0709 0.0002 -0.0008 0.0007
beta 0.0022 | 0.0000 0.0021 0.0001 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 50: -0.5681612388055 (Occupation : f = 0.5000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 p L Cm 1 d L Cm 1 f L F 1 s L F 1 p L F 1 d
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.5242 | 0.0001 0.0000 0.0015 0.5200 0.0009 0.0012 0.0002
beta 0.4758 | 0.0000 0.0002 0.0002 0.4750 0.0000 0.0001 0.0001
Gross | Cm 1 _small
-----------------------
alpha | 0.0003
beta | 0.0002
* Electronic eigenvalue no. 51: -0.5671658313030 (Occupation : f = 0.5000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.1245 | 0.1241 0.0004
beta 0.8755 | 0.8755 0.0000
* Electronic eigenvalue no. 52: -0.5637255028759 (Occupation : f = 0.5000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 d L Cm 1 f Cm 1 _small
--------------------------------------------------------------------
alpha 0.7086 | 0.0000 0.7085 0.0001
beta 0.2914 | 0.0001 0.2908 0.0003
* Electronic eigenvalue no. 53: -0.5184739266321 (Occupation : f = 0.5000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 s L Cm 1 d L Cm 1 f L F 1 s L F 1 p L F 1 d Cm 1 _small
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.4759 | 0.0002 0.0026 0.4702 0.0012 0.0014 0.0002 0.0002
beta 0.5241 | 0.0000 0.0005 0.5231 0.0000 0.0001 0.0002 0.0002
* Electronic eigenvalue no. 54: -0.5162936832339 (Occupation : f = 0.5000) m_j= -7/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.0004 | 0.0000 0.0004
beta 0.9996 | 0.9996 0.0000
* Electronic eigenvalue no. 55: -0.5126124009219 (Occupation : f = 0.5000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 p L Cm 1 d L Cm 1 f L F 1 p L F 1 d Cm 1 _small
-----------------------------------------------------------------------------------------------------------------
alpha 0.2913 | 0.0000 0.0000 0.2910 0.0000 0.0000 0.0003
beta 0.7087 | 0.0001 0.0008 0.7073 0.0002 0.0002 0.0001
* Electronic eigenvalue no. 56: -0.5125892165051 (Occupation : f = 0.5000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Cm 1 f Cm 1 _small
-----------------------------------------------------
alpha 0.8756 | 0.8755 0.0001
beta 0.1244 | 0.1241 0.0003
We note that the curium \(5f\) orbital energies are split by the molecular field, also that they are somewhat lowered. We also note that orbitals 46 - 48, dominated by fluorine \(2p\) orbitals, are almost degenerate with the curium \(5f\) orbitals, whereas orbital 50 is an almost pure curium \(7s\) orbital.