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:

  1. We calculate the curium atom.
  2. 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.
  3. 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.