Reconstructing the relativistic splitting through TD-DFT calculations

Questions
#1

Hello,

Having run some TD-DFT calculations on various silicon quantum dots, with the SOC determination enabled, I am now trying to understand the results and how they were obtained.

For reference, this is my input file for Si5H12:

$molecule
   0 1
 Si        1.360840081194     -1.360840081194      1.360840081194
 Si        1.360840081194      1.360840081194     -1.360840081194
 Si       -1.360840081194      1.360840081194      1.360840081194
 Si        0.000000000000      0.000000000000      0.000000000000
 Si       -1.360840081194     -1.360840081194     -1.360840081194
 H         2.226738765693     -0.522677587868      2.226738765693
 H         2.226738765693     -2.226738765693      0.522677587868
 H         0.522677587868     -2.226738765693      2.226738765693
 H         2.226738765693      2.226738765693     -0.522677587868
 H         0.522677587868      2.226738765693     -2.226738765693
 H         2.226738765693      0.522677587868     -2.226738765693
 H        -0.522677587868      2.226738765693      2.226738765693
 H        -2.226738765693      0.522677587868      2.226738765693
 H        -2.226738765693      2.226738765693      0.522677587868
 H        -2.226738765693     -0.522677587868     -2.226738765693
 H        -2.226738765693     -2.226738765693     -0.522677587868
 H        -0.522677587868     -2.226738765693     -2.226738765693
$end

$rem
   MEM_TOTAL            2000
   MEM_STATIC           200
   JOBTYPE              sp
   EXCHANGE             b3lyp
   BASIS                def2-svp
   SCF_ALGORITHM        diis
   MAX_SCF_CYCLES       100
   CIS_N_ROOTS          5
   CIS_SINGLETS         true
   CIS_TRIPLETS         true
   STS_MOM              true
   CALC_SOC             true
   IQMOL_FCHK           True
$end

For the SOC results, this is what I got for the values between the triplets:

=======================================================================================
                     SPIN-ORBIT COUPLING BETWEEN EXCITED TRIPLET STATES
=======================================================================================

SOC between the T1 (ms=0) state and excited triplet states (ms=1):
T2(ms=1)      -1.150200  + (0.000068i)    cm-1
T3(ms=1)      0.000054  + (0.000789i)    cm-1
T4(ms=1)      0.003656  + (0.000000i)    cm-1
T5(ms=1)      11.613783  + (0.000682i)    cm-1
T6(ms=1)      0.000580  + (0.007963i)    cm-1
SOC between the T1 (ms=0) state and excited triplet states (ms=-1):
T2(ms=-1)     1.150200  + (0.000068i)    cm-1
T3(ms=-1)     -0.000054  + (0.000789i)    cm-1
T4(ms=-1)     -0.003656  + (0.000000i)    cm-1
T5(ms=-1)     -11.613783  + (0.000682i)    cm-1
T6(ms=-1)     -0.000580  + (0.007963i)    cm-1
SOC between the T1 (ms=1) state and excited triplet states (ms=0):
T2(ms=0)      1.150200  + (0.000068i)    cm-1
T3(ms=0)      -0.000054  + (0.000789i)    cm-1
T4(ms=0)      -0.003656  + (0.000000i)    cm-1
T5(ms=0)      -11.613783  + (0.000682i)    cm-1
T6(ms=0)      -0.000580  + (0.007963i)    cm-1
SOC between the T1 (ms=1) state and excited triplet states (ms=1):
T2(ms=1)      0.000000  + (0.000077i)    cm-1
T3(ms=1)      0.000000  + (1.626626i)    cm-1
T4(ms=1)      0.000000  + (-0.002267i)    cm-1
T5(ms=1)      0.000000  + (0.000820i)    cm-1
T6(ms=1)      0.000000  + (-16.424366i)    cm-1
SOC between the T1 (ms=-1) state and excited triplet states (ms=0):
T2(ms=0)      -1.150200  + (0.000068i)    cm-1
T3(ms=0)      0.000054  + (0.000789i)    cm-1
T4(ms=0)      0.003656  + (0.000000i)    cm-1
T5(ms=0)      11.613783  + (0.000682i)    cm-1
T6(ms=0)      0.000580  + (0.007963i)    cm-1
SOC between the T1 (ms=-1) state and excited triplet states (ms=-1):
T2(ms=-1)     0.000000  + (-0.000077i)    cm-1
T3(ms=-1)     0.000000  + (-1.626626i)    cm-1
T4(ms=-1)     0.000000  + (0.002267i)    cm-1
T5(ms=-1)     0.000000  + (-0.000820i)    cm-1
T6(ms=-1)     0.000000  + (16.424366i)    cm-1
Total SOC between the T1 state and excited triplet states:
T2      2.300399    cm-1
T3      2.300398    cm-1
T4      0.007984    cm-1
T5      23.227566    cm-1
T6      23.227566    cm-1



SOC between the T2 (ms=0) state and excited triplet states (ms=1):
T3(ms=1)      -0.000068  + (-1.150199i)    cm-1
T4(ms=1)      -11.613783  + (-0.000921i)    cm-1
T5(ms=1)      0.003656  + (0.001128i)    cm-1
T6(ms=1)      0.000921  + (-11.613780i)    cm-1
SOC between the T2 (ms=0) state and excited triplet states (ms=-1):
T3(ms=-1)     0.000068  + (-1.150199i)    cm-1
T4(ms=-1)     11.613783  + (-0.000921i)    cm-1
T5(ms=-1)     -0.003656  + (0.001128i)    cm-1
T6(ms=-1)     -0.000921  + (-11.613780i)    cm-1
SOC between the T2 (ms=1) state and excited triplet states (ms=0):
T3(ms=0)      0.000068  + (-1.150199i)    cm-1
T4(ms=0)      11.613783  + (-0.000921i)    cm-1
T5(ms=0)      -0.003656  + (0.001128i)    cm-1
T6(ms=0)      -0.000921  + (-11.613780i)    cm-1
SOC between the T2 (ms=1) state and excited triplet states (ms=1):
T3(ms=1)      0.000000  + (0.001115i)    cm-1
T4(ms=1)      0.000000  + (-0.000775i)    cm-1
T5(ms=1)      0.000000  + (0.000000i)    cm-1
T6(ms=1)      0.000000  + (-0.011261i)    cm-1
SOC between the T2 (ms=-1) state and excited triplet states (ms=0):
T3(ms=0)      -0.000068  + (-1.150199i)    cm-1
T4(ms=0)      -11.613783  + (-0.000921i)    cm-1
T5(ms=0)      0.003656  + (0.001128i)    cm-1
T6(ms=0)      0.000921  + (-11.613780i)    cm-1
SOC between the T2 (ms=-1) state and excited triplet states (ms=-1):
T3(ms=-1)     0.000000  + (-0.001115i)    cm-1
T4(ms=-1)     0.000000  + (0.000775i)    cm-1
T5(ms=-1)     0.000000  + (-0.000000i)    cm-1
T6(ms=-1)     0.000000  + (0.011261i)    cm-1
Total SOC between the T2 state and excited triplet states:
T3      2.300400    cm-1
T4      23.227566    cm-1
T5      0.007652    cm-1
T6      23.227566    cm-1



SOC between the T3 (ms=0) state and excited triplet states (ms=1):
T4(ms=1)      0.000547  + (-0.004306i)    cm-1
T5(ms=1)      -0.000682  + (11.613782i)    cm-1
T6(ms=1)      -0.000000  + (0.001128i)    cm-1
SOC between the T3 (ms=0) state and excited triplet states (ms=-1):
T4(ms=-1)     -0.000547  + (-0.004306i)    cm-1
T5(ms=-1)     0.000682  + (11.613782i)    cm-1
T6(ms=-1)     0.000000  + (0.001128i)    cm-1
SOC between the T3 (ms=1) state and excited triplet states (ms=0):
T4(ms=0)      -0.000547  + (-0.004306i)    cm-1
T5(ms=0)      0.000682  + (11.613782i)    cm-1
T6(ms=0)      0.000000  + (0.001128i)    cm-1
SOC between the T3 (ms=1) state and excited triplet states (ms=1):
T4(ms=1)      0.000000  + (-16.424369i)    cm-1
T5(ms=1)      0.000000  + (-0.006090i)    cm-1
T6(ms=1)      0.000000  + (0.002267i)    cm-1
SOC between the T3 (ms=-1) state and excited triplet states (ms=0):
T4(ms=0)      0.000547  + (-0.004306i)    cm-1
T5(ms=0)      -0.000682  + (11.613782i)    cm-1
T6(ms=0)      -0.000000  + (0.001128i)    cm-1
SOC between the T3 (ms=-1) state and excited triplet states (ms=-1):
T4(ms=-1)     0.000000  + (16.424369i)    cm-1
T5(ms=-1)     0.000000  + (0.006090i)    cm-1
T6(ms=-1)     0.000000  + (-0.002267i)    cm-1
Total SOC between the T3 state and excited triplet states:
T4      23.227567    cm-1
T5      23.227567    cm-1
T6      0.003920    cm-1



SOC between the T4 (ms=0) state and excited triplet states (ms=1):
T5(ms=1)      4.568950  + (-0.000362i)    cm-1
T6(ms=1)      0.000228  + (-0.001694i)    cm-1
SOC between the T4 (ms=0) state and excited triplet states (ms=-1):
T5(ms=-1)     -4.568950  + (-0.000362i)    cm-1
T6(ms=-1)     -0.000228  + (-0.001694i)    cm-1
SOC between the T4 (ms=1) state and excited triplet states (ms=0):
T5(ms=0)      -4.568950  + (-0.000362i)    cm-1
T6(ms=0)      -0.000228  + (-0.001694i)    cm-1
SOC between the T4 (ms=1) state and excited triplet states (ms=1):
T5(ms=1)      0.000000  + (-0.000323i)    cm-1
T6(ms=1)      0.000000  + (6.461469i)    cm-1
SOC between the T4 (ms=-1) state and excited triplet states (ms=0):
T5(ms=0)      4.568950  + (-0.000362i)    cm-1
T6(ms=0)      0.000228  + (-0.001694i)    cm-1
SOC between the T4 (ms=-1) state and excited triplet states (ms=-1):
T5(ms=-1)     0.000000  + (0.000323i)    cm-1
T6(ms=-1)     0.000000  + (-6.461469i)    cm-1
Total SOC between the T4 state and excited triplet states:
T5      9.137901    cm-1
T6      9.137898    cm-1



SOC between the T5 (ms=0) state and excited triplet states (ms=1):
T6(ms=1)      0.000362  + (4.568950i)    cm-1
SOC between the T5 (ms=0) state and excited triplet states (ms=-1):
T6(ms=-1)     -0.000362  + (4.568950i)    cm-1
SOC between the T5 (ms=1) state and excited triplet states (ms=0):
T6(ms=0)      -0.000362  + (4.568950i)    cm-1
SOC between the T5 (ms=1) state and excited triplet states (ms=1):
T6(ms=1)      0.000000  + (0.002396i)    cm-1
SOC between the T5 (ms=-1) state and excited triplet states (ms=0):
T6(ms=0)      0.000362  + (4.568950i)    cm-1
SOC between the T5 (ms=-1) state and excited triplet states (ms=-1):
T6(ms=-1)     0.000000  + (-0.002396i)    cm-1
Total SOC between the T5 state and excited triplet states:
T6      9.137900    cm-1

And the values between singlets and triplets:

=======================================================================================
         SPIN-ORBIT COUPLING BETWEEN EXCITED SINGLET STATES AND TRIPLET STATES
=======================================================================================

SOC between the S1 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  0.000079i    cm-1
T2(ms=0)      0.000000  +  0.000000i    cm-1
T3(ms=0)      0.000000  +  0.000546i    cm-1
T4(ms=0)      0.000000  +  -0.000037i    cm-1
T5(ms=0)      0.000000  +  -0.000000i    cm-1
T6(ms=0)      0.000000  +  0.000256i    cm-1
SOC between the S1 state and excited triplet states (ms=1):
T1(ms=1)      (-25.797996)  + (0.001515i)    cm-1
T2(ms=1)      (-0.017301)  + (0.001272i)    cm-1
T3(ms=1)      (0.001516)  + (25.798001i)    cm-1
T4(ms=1)      (-12.103680)  + (-0.000959i)    cm-1
T5(ms=1)      (-0.004307)  + (0.000579i)    cm-1
T6(ms=1)      (0.000959)  + (-12.103681i)    cm-1
SOC between the S1 state and excited triplet states (ms=-1):
T1(ms=-1)    -(-25.797996)  + (0.001515i)    cm-1
T2(ms=-1)    -(-0.017301)  + (0.001272i)    cm-1
T3(ms=-1)    -(0.001516)  + (25.798001i)    cm-1
T4(ms=-1)    -(-12.103680)  + (-0.000959i)    cm-1
T5(ms=-1)    -(-0.004307)  + (0.000579i)    cm-1
T6(ms=-1)    -(0.000959)  + (-12.103681i)    cm-1
Total SOC between the S1 state and excited triplet states:
T1      36.483875    cm-1
T2      0.024534    cm-1
T3      36.483883    cm-1
T4      17.117189    cm-1
T5      0.006146    cm-1
T6      17.117190    cm-1



SOC between the S2 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  0.002180i    cm-1
T2(ms=0)      0.000000  +  0.001720i    cm-1
T3(ms=0)      0.000000  +  36.483886i    cm-1
T4(ms=0)      0.000000  +  0.001340i    cm-1
T5(ms=0)      0.000000  +  -0.000855i    cm-1
T6(ms=0)      0.000000  +  17.117188i    cm-1
SOC between the S2 state and excited triplet states (ms=1):
T1(ms=1)      (-0.017301)  + (-0.000000i)    cm-1
T2(ms=1)      (25.797995)  + (0.000026i)    cm-1
T3(ms=1)      (-0.001215)  + (-0.000386i)    cm-1
T4(ms=1)      (-0.004307)  + (0.000000i)    cm-1
T5(ms=1)      (12.103679)  + (-0.000012i)    cm-1
T6(ms=1)      (0.000605)  + (0.000181i)    cm-1
SOC between the S2 state and excited triplet states (ms=-1):
T1(ms=-1)    -(-0.017301)  + (-0.000000i)    cm-1
T2(ms=-1)    -(25.797995)  + (0.000026i)    cm-1
T3(ms=-1)    -(-0.001215)  + (-0.000386i)    cm-1
T4(ms=-1)    -(-0.004307)  + (0.000000i)    cm-1
T5(ms=-1)    -(12.103679)  + (-0.000012i)    cm-1
T6(ms=-1)    -(0.000605)  + (0.000181i)    cm-1
Total SOC between the S2 state and excited triplet states:
T1      0.024565    cm-1
T2      36.483875    cm-1
T3      36.483886    cm-1
T4      0.006236    cm-1
T5      17.117188    cm-1
T6      17.117188    cm-1



SOC between the S3 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  -36.483878i    cm-1
T2(ms=0)      0.000000  +  -0.025014i    cm-1
T3(ms=0)      0.000000  +  0.002181i    cm-1
T4(ms=0)      0.000000  +  17.117188i    cm-1
T5(ms=0)      0.000000  +  0.006347i    cm-1
T6(ms=0)      0.000000  +  -0.001339i    cm-1
SOC between the S3 state and excited triplet states (ms=1):
T1(ms=1)      (-0.000056)  + (0.017688i)    cm-1
T2(ms=1)      (0.000026)  + (-25.797995i)    cm-1
T3(ms=1)      (0.000000)  + (0.001271i)    cm-1
T4(ms=1)      (-0.000026)  + (-0.004488i)    cm-1
T5(ms=1)      (0.000012)  + (12.103680i)    cm-1
T6(ms=1)      (0.000000)  + (0.000579i)    cm-1
SOC between the S3 state and excited triplet states (ms=-1):
T1(ms=-1)    -(-0.000056)  + (0.017688i)    cm-1
T2(ms=-1)    -(0.000026)  + (-25.797995i)    cm-1
T3(ms=-1)    -(0.000000)  + (0.001271i)    cm-1
T4(ms=-1)    -(-0.000026)  + (-0.004488i)    cm-1
T5(ms=-1)    -(0.000012)  + (12.103680i)    cm-1
T6(ms=-1)    -(0.000000)  + (0.000579i)    cm-1
Total SOC between the S3 state and excited triplet states:
T1      36.483886    cm-1
T2      36.483883    cm-1
T3      0.002826    cm-1
T4      17.117189    cm-1
T5      17.117189    cm-1
T6      0.001570    cm-1



SOC between the S4 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  -28.820004i    cm-1
T2(ms=0)      0.000000  +  -0.019760i    cm-1
T3(ms=0)      0.000000  +  0.001742i    cm-1
T4(ms=0)      0.000000  +  6.426291i    cm-1
T5(ms=0)      0.000000  +  0.002383i    cm-1
T6(ms=0)      0.000000  +  -0.000498i    cm-1
SOC between the S4 state and excited triplet states (ms=1):
T1(ms=1)      (0.000133)  + (-0.013972i)    cm-1
T2(ms=1)      (-0.000034)  + (20.378822i)    cm-1
T3(ms=1)      (-0.000000)  + (-0.000827i)    cm-1
T4(ms=1)      (0.000030)  + (0.001685i)    cm-1
T5(ms=1)      (-0.000008)  + (-4.544074i)    cm-1
T6(ms=1)      (-0.000000)  + (-0.000257i)    cm-1
SOC between the S4 state and excited triplet states (ms=-1):
T1(ms=-1)    -(0.000133)  + (-0.013972i)    cm-1
T2(ms=-1)    -(-0.000034)  + (20.378822i)    cm-1
T3(ms=-1)    -(-0.000000)  + (-0.000827i)    cm-1
T4(ms=-1)    -(0.000030)  + (0.001685i)    cm-1
T5(ms=-1)    -(-0.000008)  + (-4.544074i)    cm-1
T6(ms=-1)    -(-0.000000)  + (-0.000257i)    cm-1
Total SOC between the S4 state and excited triplet states:
T1      28.820011    cm-1
T2      28.820013    cm-1
T3      0.002099    cm-1
T4      6.426292    cm-1
T5      6.426292    cm-1
T6      0.000617    cm-1



SOC between the S5 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  -0.001742i    cm-1
T2(ms=0)      0.000000  +  -0.001359i    cm-1
T3(ms=0)      0.000000  +  -28.820012i    cm-1
T4(ms=0)      0.000000  +  -0.000499i    cm-1
T5(ms=0)      0.000000  +  0.000321i    cm-1
T6(ms=0)      0.000000  +  -6.426292i    cm-1
SOC between the S5 state and excited triplet states (ms=1):
T1(ms=1)      (-0.014498)  + (-0.000000i)    cm-1
T2(ms=1)      (20.378821)  + (0.000034i)    cm-1
T3(ms=1)      (-0.000960)  + (-0.000525i)    cm-1
T4(ms=1)      (-0.001802)  + (0.000000i)    cm-1
T5(ms=1)      (4.544074)  + (-0.000008i)    cm-1
T6(ms=1)      (0.000227)  + (0.000117i)    cm-1
SOC between the S5 state and excited triplet states (ms=-1):
T1(ms=-1)    -(-0.014498)  + (-0.000000i)    cm-1
T2(ms=-1)    -(20.378821)  + (0.000034i)    cm-1
T3(ms=-1)    -(-0.000960)  + (-0.000525i)    cm-1
T4(ms=-1)    -(-0.001802)  + (0.000000i)    cm-1
T5(ms=-1)    -(4.544074)  + (-0.000008i)    cm-1
T6(ms=-1)    -(0.000227)  + (0.000117i)    cm-1
Total SOC between the S5 state and excited triplet states:
T1      0.020576    cm-1
T2      28.820005    cm-1
T3      28.820012    cm-1
T4      0.002597    cm-1
T5      6.426291    cm-1
T6      6.426292    cm-1



SOC between the S6 state and excited triplet states (ms=0):
T1(ms=0)      0.000000  +  -0.000188i    cm-1
T2(ms=0)      0.000000  +  -0.000000i    cm-1
T3(ms=0)      0.000000  +  0.000743i    cm-1
T4(ms=0)      0.000000  +  0.000042i    cm-1
T5(ms=0)      0.000000  +  0.000000i    cm-1
T6(ms=0)      0.000000  +  0.000166i    cm-1
SOC between the S6 state and excited triplet states (ms=1):
T1(ms=1)      (-20.378820)  + (-0.001197i)    cm-1
T2(ms=1)      (-0.014497)  + (-0.000828i)    cm-1
T3(ms=1)      (0.001198)  + (-20.378826i)    cm-1
T4(ms=1)      (-4.544075)  + (0.000360i)    cm-1
T5(ms=1)      (-0.001802)  + (-0.000257i)    cm-1
T6(ms=1)      (0.000360)  + (4.544075i)    cm-1
SOC between the S6 state and excited triplet states (ms=-1):
T1(ms=-1)    -(-20.378820)  + (-0.001197i)    cm-1
T2(ms=-1)    -(-0.014497)  + (-0.000828i)    cm-1
T3(ms=-1)    -(0.001198)  + (-20.378826i)    cm-1
T4(ms=-1)    -(-4.544075)  + (0.000360i)    cm-1
T5(ms=-1)    -(-0.001802)  + (-0.000257i)    cm-1
T6(ms=-1)    -(0.000360)  + (4.544075i)    cm-1
Total SOC between the S6 state and excited triplet states:
T1      28.820004    cm-1
T2      0.020536    cm-1
T3      28.820012    cm-1
T4      6.426292    cm-1
T5      0.002574    cm-1
T6      6.426293    cm-1

I’m not sure I understand how we got those results though. It looks like we only obtain one spin-orbit matrix, aren’t we supposed to obtain multiple ones? Is it the average value? How can we reconstruct the relativistic splitting of the electronic levels based on those data? I’ve read the 7.3.6 section of the manual but I still don’t understand it very well.

Thank you very much for your help!

Best regards!

Iacobellis Nicolas, Ph.D. Student

#2

Can I ask why you are expecting multiple spin-orbit matrices?

#3

We were expecting one matrix for each symmetry resulting from the coupling of the spatial symmetry and the spin symmetry (double groups). So we should obtain as many matrices as they are symmetries in the Td point group. Or it could be one big matrix that is block diagonal with zeroes between different symmetries.

In this case though, it looks like we have as many relativistic states as non-relavistic ones. Did we miss something?

#4

So, you are expecting multiple matrices which correspond to the SOC of the states belong to the same spatial symmetries, or a large matrix which is block-diagonal according to the spatial symmetries, right? However, there are no spatial symmetry information in your output, so I’m not sure whether those singlets/triplets belong to the same spatial symmetry or not. Moreover, why did you expect different no. of “relativistic” states from no. of “non-relativistic” states? In the output you can see the different Ms components of the triplets have different SOC values, while if without SOC, these components are degenerate in the non-relativistic case. Not sure if I understand your question correctly.

#5

This is the complete output file for this calculation: https://file.io/JUtLo6tnUkuk
It is too long to just be pasted here and I can’t upload anything other than pictures directly through this site.

Our goal is to determinate the relativistic splitting of the electronic levels, so the number of relativistic states should be higher than the non-relativistic ones. If it is possible, how can we reconstruct the splitting of the energy levels based on those data?