Mr. Gao,

The SO is large in gas-phase, but in condense phase there is a quencing of the orbital angular moment, and thereby, its (meaning SO) is not important.

 Hi Ivanova

Thanks for your answer. So it is right that the Cu would has a large SO splitting in gas-phase. Beacuse I am doing some researches about the absorption of metal atoms and so the SO splitting becomes important.

I wonder, how can an orbitally S state have a spin-orbit splitting at all ?

Mr. Gao,

From your additional comments, it can be deduced that you most probably compute energetics of adsorbed species (Most probably thermodynamics?!).

However, from the well known Russell-Saunders' methodology is well known, too, that the SO energy can be given by:

ESO = LSL.(1/2).(J(J+1)-S(S+1)-L(L+1))

So, that at S = 0 (J=L), ESO is '0'.

Therefore, for GS computations both in solution and/or gas-phase, you do not need SO. Or you can obtain the energetics of the system (adsorbent)(gas) + Cu-surface = adsorbent-Cu-surface without to take into account for SO of 'adsorbent-Cu-surface'.

If we use the equation for total energy:

E = EKin + Epot + EInt,

and approximate the EInt (electromagnetic interaction) with SO interaction (This approximation is also known), this means that if EKin >> EInt, then EInt term can be neglected.

On the other hand, for the kinetic energy we use rotational kinetic energy given by:

(1/2).I.(omega)2 = l2/2I,

where l2 is orbital angular moment. The large angular moment causes for large rotational kinetic energy. Thus, EInt (above) can be neglected. Cu has large angula moment. Taking into consideration that the magnetic moment is proportional to the orbital moment with a sing '-', the large orbital angula moments in Cu, agrees with the known data about very small magnetic moments of its compounds, in general.

In this context, you could pay attention to Ref. 1, for example, studying magnetic properties of Cu-clusters with different amount Cu (Table 4). With increasing in the Cu-amount there has been found decreasing in local magnetic moments up to 0.02 bm.

Ref. 1. D.Die, Z. Zheng, L. Zhao, Q. Zhu, Z. Zhao, Sci. Reports 6 (2016) 31978; DOI: 10.1038/sper31978.

Or, even studying adsorption processes onto Cu surfaces, SO is not important in your case.

SO is important, rather to 3rd raw d-block.

For example, SO for HgS is 156.1 cm-1 (or ca. 0.019 eV, below).

HgS:

*****************************

Spin-Orbit coupling program.

*****************************

Number of configs= 4

1st state is 1

2nd state is 2

Transition Spin Density Matrix

1 2

1 -0.110217D-11 0.141120D+01

2 0.922237D-01 0.110217D-11

Magnitude in x-direction= 156.1 cm-1

Magnitude in y-direction= 0.0 cm-1

Magnitude in z-direction= 0.0 cm-1

Total magnitude= 156.1 cm-1

MCSCF converged.

By contrast, the analysis of NiS, which is 1st row d-block shows:

*****************************

Spin-Orbit coupling program.

*****************************

Number of configs= 4

1st state is 1

2nd state is 2

Transition Spin Density Matrix

1 2

1 -0.100000D+01 -0.148723D-09

2 -0.154968D-09 0.100000D+01

Magnitude in x-direction= 0.0 cm-1

Magnitude in y-direction= 0.0 cm-1

Magnitude in z-direction= 0.0 cm-1

Total magnitude= 0.0 cm-1

The latter computational result agrees with the known rule stated above, too.

Hi Fiorentini

Yes, the S orbit should not have a SOC for L=0. In the calculation result I posted in the question, it is clear that the d orbit has a spin orbit splitting of 40 meV, which is reasonable. For the s orbit, it is more like a spin polarized state because there is only one electron in s orbit.  As my understanding, the s orbit is a sphere in space and so it is isotropy in every direction.  When I am doing the SOC calculation, it is actually doing a collinear magnetic calculation (spin polarized) instead of non-collinear calculation. So the large energy splitting of 0.8 eV may be contributed to the splitting of spin-up and spin-down.  However, the situation would change when the Cu atom adsorbed on graphene, it is anisotropy for Cu 4s orbit. And I also get a large energy splitting for S orbit. So when Cu atom is adsorbed on graphene, there are both spin polarization and spin-orbit coupling in the S orbit. 

Hi Ivanova

Thanks very much for your detailed explanation which is very useful for me.

Beacuse I need to calculate the adsorption energy of Cu atom on graphene.  E(adsorption)=E[Cu/graphene]-E[Cu]-E[graphene]. You are correct that for Cu atom the E(soc)=0, but I believe that one Cu atom would be Spin polarized with M=1 uB. And the lager energy splitting is due to the spin polarized in s orbit. My system only contains one Cu atom and graphene, so the magnetic moment of Cu is significant. Do I need to take consideration of the spin polarization when calculating the energy of Cu atom to get the adsorption energy? And do the spin polarization affect the adsorption of Cu on graphene?

Thanks.

Mr. Gao,

Shall you maximize the orbital angular moment or not, depends on the average orbital angular moment of the system. For open shell systems you need it.

If you compute Cu0, for example, then you should add polarization function, because of it is an open shell system (El. config. 4s13d10). Please pay attention to the NBO analysis, orbital and kinetic energies and laplacian electron densities of isolated Cu0 in gas-phase:

Cu0:

NATURAL POPULATIONS: Natural atomic orbital occupancies

NAO Atom No lang Type(AO) Occupancy

--------------------------------------------

1 Cu 1 S Cor( 3S) 2.00000

2 Cu 1 S Val( 4S) 1.00000

3 Cu 1 S Ryd( 5S) 0.00000

4 Cu 1 S Ryd(10S) 0.00000

5 Cu 1 S Ryd( 9S) 0.00000

6 Cu 1 S Ryd( 8S) 0.00000

7 Cu 1 S Ryd( 7S) 0.00000

8 Cu 1 S Ryd( 6S) 0.00000

9 Cu 1 S Ryd(11S) 0.00000

10 Cu 1 px Cor( 3p) 2.00000

11 Cu 1 px Val( 4p) 0.00000

12 Cu 1 px Ryd( 7p) 0.00000

13 Cu 1 px Ryd( 6p) 0.00000

14 Cu 1 px Ryd( 5p) 0.00000

15 Cu 1 py Cor( 3p) 2.00000

16 Cu 1 py Val( 4p) 0.00000

17 Cu 1 py Ryd( 7p) 0.00000

18 Cu 1 py Ryd( 6p) 0.00000

19 Cu 1 py Ryd( 5p) 0.00000

20 Cu 1 pz Cor( 3p) 2.00000

21 Cu 1 pz Val( 4p) 0.00000

22 Cu 1 pz Ryd( 7p) 0.00000

23 Cu 1 pz Ryd( 6p) 0.00000

24 Cu 1 pz Ryd( 5p) 0.00000

25 Cu 1 dxy Val( 3d) 1.99973

26 Cu 1 dxy Ryd( 4d) 0.00027

27 Cu 1 dxy Ryd( 5d) 0.00000

28 Cu 1 dxz Val( 3d) 1.99973

29 Cu 1 dxz Ryd( 4d) 0.00027

30 Cu 1 dxz Ryd( 5d) 0.00000

31 Cu 1 dyz Val( 3d) 1.99973

32 Cu 1 dyz Ryd( 4d) 0.00027

33 Cu 1 dyz Ryd( 5d) 0.00000

34 Cu 1 dx2y2 Val( 3d) 1.99973

35 Cu 1 dx2y2 Ryd( 4d) 0.00027

36 Cu 1 dx2y2 Ryd( 5d) 0.00000

37 Cu 1 dz2 Val( 3d) 1.99973

38 Cu 1 dz2 Ryd( 4d) 0.00027

39 Cu 1 dz2 Ryd( 5d) 0.00000

..............................................

Atom No Natural Electron Configuration

------------------------------------------------------

Cu 1 [core]4S( 1.00)3d(10.00)

.............

NATURAL POPULATIONS: Natural atomic orbital occupancies

NAO Atom No lang Type(AO) Occupancy Energy

----------------------------------------------------------

1 Cu 1 S Val( 3S) 1.00000 -4.47885

2 Cu 1 S Cor( 4S) 1.00000 -0.20230

3 Cu 1 S Ryd( 5S) 0.00000 0.76984

4 Cu 1 S Ryd( 7S) 0.00000 1.97176

5 Cu 1 S Ryd( 8S) 0.00000 2.06804

6 Cu 1 S Ryd(10S) 0.00000 72.42601

7 Cu 1 S Ryd( 6S) 0.00000 1.43812

8 Cu 1 S Ryd( 9S) 0.00000 7.21097

9 Cu 1 S Ryd(11S) 0.00000 126.65919

10 Cu 1 px Cor( 3p) 1.00000 -2.77486

11 Cu 1 px Val( 4p) 0.00000 0.45075

12 Cu 1 px Ryd( 7p) 0.00000 54.37854

13 Cu 1 px Ryd( 6p) 0.00000 3.16306

14 Cu 1 px Ryd( 5p) 0.00000 2.19478

15 Cu 1 py Cor( 3p) 1.00000 -2.77486

16 Cu 1 py Val( 4p) 0.00000 0.45075

17 Cu 1 py Ryd( 7p) 0.00000 54.37854

18 Cu 1 py Ryd( 6p) 0.00000 3.16306

19 Cu 1 py Ryd( 5p) 0.00000 2.19478

20 Cu 1 pz Cor( 3p) 1.00000 -2.77486

21 Cu 1 pz Val( 4p) 0.00000 0.45075

22 Cu 1 pz Ryd( 7p) 0.00000 54.37854

23 Cu 1 pz Ryd( 6p) 0.00000 3.16306

24 Cu 1 pz Ryd( 5p) 0.00000 2.19478

25 Cu 1 dxy Val( 3d) 0.99986 -0.25020

26 Cu 1 dxy Ryd( 4d) 0.00014 0.40732

27 Cu 1 dxy Ryd( 5d) 0.00000 1.02279

28 Cu 1 dxz Val( 3d) 0.99986 -0.25020

29 Cu 1 dxz Ryd( 4d) 0.00014 0.40732

30 Cu 1 dxz Ryd( 5d) 0.00000 1.02279

31 Cu 1 dyz Val( 3d) 0.99986 -0.25020

32 Cu 1 dyz Ryd( 4d) 0.00014 0.40732

33 Cu 1 dyz Ryd( 5d) 0.00000 1.02279

34 Cu 1 dx2y2 Val( 3d) 0.99986 -0.25020

35 Cu 1 dx2y2 Ryd( 4d) 0.00014 0.40732

36 Cu 1 dx2y2 Ryd( 5d) 0.00000 1.02279

37 Cu 1 dz2 Val( 3d) 0.99986 -0.25020

38 Cu 1 dz2 Ryd( 4d) 0.00014 0.40732

39 Cu 1 dz2 Ryd( 5d) 0.00000 1.02279

.......................

(Occupancy) Bond orbital/ Coefficients/ Hybrids

---------------------------------------------------------------------------------

1. (1.00000) CR ( 1)Cu 1 s(100.00%)

0.0000 1.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

2. (1.00000) CR ( 2)Cu 1 s( 0.00%)p 1.00(100.00%)

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

3. (1.00000) CR ( 3)Cu 1 s( 0.00%)p 1.00(100.00%)

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

4. (1.00000) CR ( 4)Cu 1 s( 0.00%)p 1.00(100.00%)

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

..................

Orbital energies and kinetic energies (alpha):

1 2

1 (A1G)--O -4.478851 3.017753

2 (T1U)--O -2.774866 5.113570

3 (T1U)--O -2.774866 5.113570

4 (T1U)--O -2.774866 5.113570

5 (EG)--O -0.250288 9.213664

6 (T2G)--O -0.250288 9.213664

7 (T2G)--O -0.250288 9.213664

8 (T2G)--O -0.250288 9.213664

9 (EG)--O -0.250288 9.213664

10 (A1G)--O -0.202296 0.427650

11 (T1U)--V -0.015183 0.207635

12 (T1U)--V -0.015183 0.207635

13 (T1U)--V -0.015183 0.207635

14 (A1G)--V 0.005533 0.053397

15 (A1G)--V 0.101260 0.256805

16 (T1U)--V 0.145578 0.556666

17 (T1U)--V 0.145578 0.556666

18 (T1U)--V 0.145578 0.556666

19 (T2G)--V 0.216502 1.002437

20 (T2G)--V 0.216502 1.002437

21 (T2G)--V 0.216502 1.002437

22 (EG)--V 0.216502 1.002437

23 (EG)--V 0.216502 1.002437

24 (A1G)--V 0.384466 0.765202

25 (A1G)--V 1.115046 1.660678

26 (T2G)--V 1.213696 5.544616

27 (T2G)--V 1.213696 5.544616

28 (T2G)--V 1.213696 5.544616

29 (EG)--V 1.213696 5.544616

30 (EG)--V 1.213696 5.544616

31 (T1U)--V 2.617220 6.971489

32 (T1U)--V 2.617220 6.971489

33 (T1U)--V 2.617220 6.971489

34 (A1G)--V 3.779423 7.095538

35 (A1G)--V 50.484656 32.440142

36 (T1U)--V 57.439511 48.527530

37 (T1U)--V 57.439511 48.527530

38 (T1U)--V 57.439511 48.527530

39 (A1G)--V 156.673543 64.977034

Orbital energies and kinetic energies (beta):

1 2

1 (A1G)--O -4.480490 3.018799

2 (T1U)--O -2.777266 5.123052

3 (T1U)--O -2.777266 5.123052

4 (T1U)--O -2.777266 5.123052

5 (EG)--O -0.247110 9.425712

6 (T2G)--O -0.247110 9.425712

7 (T2G)--O -0.247110 9.425712

8 (T2G)--O -0.247110 9.425712

9 (EG)--O -0.247110 9.425712

10 (A1G)--V -0.110452 0.386313

11 (A1G)--V 0.012988 0.058124

12 (T1U)--V 0.014480 0.158841

13 (T1U)--V 0.014480 0.158841

14 (T1U)--V 0.014480 0.158841

15 (A1G)--V 0.117557 0.277530

16 (T1U)--V 0.177108 0.605484

17 (T1U)--V 0.177108 0.605484

18 (T1U)--V 0.177108 0.605484

19 (T2G)--V 0.247136 0.898230

20 (T2G)--V 0.247136 0.898230

21 (T2G)--V 0.247136 0.898230

22 (EG)--V 0.247136 0.898230

23 (EG)--V 0.247136 0.898230

24 (A1G)--V 0.407139 0.780352

25 (A1G)--V 1.142383 1.669337

26 (T2G)--V 1.240090 5.436775

27 (T2G)--V 1.240090 5.436775

28 (T2G)--V 1.240090 5.436775

29 (EG)--V 1.240090 5.436775

30 (EG)--V 1.240090 5.436775

31 (T1U)--V 2.628135 6.961939

32 (T1U)--V 2.628135 6.961939

33 (T1U)--V 2.628135 6.961939

34 (A1G)--V 3.792397 7.086395

35 (A1G)--V 50.479622 32.439075

36 (T1U)--V 57.431125 48.527572

37 (T1U)--V 57.431125 48.527572

38 (T1U)--V 57.431125 48.527572

39 (A1G)--V 156.669738 64.978274

Total kinetic energy from orbitals= 1.303709493783D+02

....................

Gross orbital populations:

Total Alpha Beta Spin

1 1 Cu 1S -0.28102 -0.14410 -0.13693 -0.00717

2 2S 1.37178 0.69828 0.67351 0.02477

3 3S 0.85508 0.42008 0.43500 -0.01493

4 4S 0.58953 0.59312 -0.00359 0.59671

5 5S 0.32238 0.32254 -0.00015 0.32269

6 6S 0.00233 0.00233 0.00000 0.00233

7 7PX -0.06239 -0.03116 -0.03123 0.00008

8 7PY -0.06239 -0.03116 -0.03123 0.00008

9 7PZ -0.06239 -0.03116 -0.03123 0.00008

10 8PX 1.41780 0.70813 0.70966 -0.00153

11 8PY 1.41780 0.70813 0.70966 -0.00153

12 8PZ 1.41780 0.70813 0.70966 -0.00153

13 9PX 0.63836 0.31971 0.31865 0.00106

14 9PY 0.63836 0.31971 0.31865 0.00106

15 9PZ 0.63836 0.31971 0.31865 0.00106

16 10PX 0.00654 0.00348 0.00307 0.00041

17 10PY 0.00654 0.00348 0.00307 0.00041

18 10PZ 0.00654 0.00348 0.00307 0.00041

19 11PX -0.00031 -0.00017 -0.00015 -0.00002

20 11PY -0.00031 -0.00017 -0.00015 -0.00002

21 11PZ -0.00031 -0.00017 -0.00015 -0.00002

22 12XX 1.07353 0.53128 0.54225 -0.01097

23 12YY 1.07353 0.53128 0.54225 -0.01097

24 12ZZ 1.07353 0.53128 0.54225 -0.01097

25 12XY 1.58353 0.78366 0.79987 -0.01620

26 12XZ 1.58353 0.78366 0.79987 -0.01620

27 12YZ 1.58353 0.78366 0.79987 -0.01620

28 13XX 0.27223 0.15027 0.12196 0.02831

29 13YY 0.27223 0.15027 0.12196 0.02831

30 13ZZ 0.27223 0.15027 0.12196 0.02831

31 13XY 0.37088 0.19043 0.18045 0.00999

32 13XZ 0.37088 0.19043 0.18045 0.00999

33 13YZ 0.37088 0.19043 0.18045 0.00999

34 14XX 0.03420 0.02103 0.01317 0.00786

35 14YY 0.03420 0.02103 0.01317 0.00786

36 14ZZ 0.03420 0.02103 0.01317 0.00786

37 14XY 0.04559 0.02590 0.01969 0.00622

38 14XZ 0.04559 0.02590 0.01969 0.00622

39 14YZ 0.04559 0.02590 0.01969 0.00622

...........................................

For ensembles Cu0/(C)n you should account for the electronic configurations and the number of electrons.

Mr. Gao,

In addition you could pay attention to the data about SO of Cu0 by LMTO/LSDA-U (with GGA: Perdew 91, kappas=2):

SO-energy = (-0.6886613E-01, 0.5248106E-10)

up-up = (-0.3443370E-01,-0.1169298E-15)

dn-dn = (-0.3443243E-01, 0.1957807E-15)

up-dn = (-0.9739538E-01, 0.2624051E-10)

dn-up = ( 0.9739538E-01, 0.2624047E-10)

As we could expect, there is a weak effect on the total energy of the system, and, respectively, to the adsorption energetics.

Because of the magnitude of the total enrgy is:

TOTAL ENERGY =-1321.95318691486.