For this example using relativistic density functional theory, let's stick to our trusty friend water. Specifically, we will be using the eXact 2-Component (X2C) Hamiltonian1, which includes one-electron spin-orbit coupling and scalar relativistic effects.
The 2-component framework follows that of the Generalized Hartree-Fock (GHF) method, where the quantum number Sz is not restricted, meaning that electronic spins are allowed to rotate freely between the \alpha and \beta manifolds.
Input
The following input is for a water molecule using the 6-31g(d) basis, and the B3LYP DFT functional with the X2C relativistic Hamiltonian. The reference keyword is therefore reference = X2CB3LYP, and since we are using X2C it will default to complex arithmetic, which is equivalent to reference = Complex X2CB3LYP. Let's name this file water_x2c.inp:
[Molecule]
charge = 0
mult = 1
geom:
O 0.0 -0.076 0.0
H 0.867 0.601 0.0
H -0.867 0.601 0.0
[QM]
reference = X2CB3LYP
job = SCF
[BASIS]
basis = 6-31g(d)SCF Results
After running this calculation using
./chronusq water_x2c.inpwe can examine the output file water_x2c.out. Specifically let's look at the ground state SCF results:
--------------------------------------------------------------------------------
SCF Iteration Energy (Eh) ΔE (Eh) |ΔP(S)| |ΔP(M)|
------------- ----------- ------- ------- -------
SCFIt: 0 -77.6512427603
SCFIt: 1 -74.4256509007 3.2255919e+00 5.7332715e-02 1.7344131e-04
SCFIt: 2 -78.7846742155 -4.3590233e+00 6.0299790e-02 1.5504369e-04
SCFIt: 3 -75.5333069080 3.2513673e+00 2.1901754e-02 5.8227433e-05
SCFIt: 4 -76.2736425358 -7.4033563e-01 4.2499336e-03 1.4200872e-05
SCFIt: 5 -76.5007418288 -2.2709929e-01 1.7744060e-03 4.5621012e-06
SCFIt: 6 -76.4325517674 6.8190061e-02 7.8868972e-06 1.9850302e-07
*** Damping Disabled - Energy Difference Fell Below 1.0000000e-03 ***
SCFIt: 7 -76.4326197211 -6.7953680e-05 2.3745738e-06 3.1878592e-08
SCFIt: 8 -76.4325893507 3.0370435e-05 9.3499761e-07 1.3074593e-08
SCFIt: 9 -76.4326364233 -4.7072577e-05 1.9858860e-06 4.5099070e-09
SCFIt: 10 -76.4325777509 5.8672362e-05 5.0310472e-07 1.3407474e-09
SCFIt: 11 -76.4324696558 1.0809511e-04 2.2462960e-06 7.2447879e-09
SCFIt: 12 -76.4325724945 -1.0283877e-04 3.6853204e-07 8.5337007e-10
SCFIt: 13 -76.4325383474 3.4147124e-05 5.0119513e-07 1.6182022e-09
SCFIt: 14 -76.4325258659 1.2481570e-05 8.2672785e-07 3.5835837e-09
SCFIt: 15 -76.4325610736 -3.5207771e-05 8.0050097e-08 6.2001542e-10
SCFIt: 16 -76.4325586530 2.4205876e-06 1.6222970e-08 9.5563342e-11
SCFIt: 17 -76.4325580395 6.1350738e-07 1.5629323e-10 2.6631709e-12
SCFIt: 18 -76.4325580334 6.0927761e-09 1.5553722e-13 6.6264729e-13
SCFIt: 19 -76.4325580334 -5.6132876e-12 2.8857653e-14 5.6196063e-14
SCF Completed: E(C-X2C-B3LYP) = -76.4325580334 Eh after 19 SCF Iterations
================================================================================
SCF Results:
================================================================================
Orbital Eigenenergies / Eh
--------------------------------------------------------------------------------
Occupied:
-1.9171e+01 -1.9171e+01 -9.5084e-01 -9.5084e-01 -4.7269e-01
-4.7269e-01 -3.6280e-01 -3.6280e-01 -2.8289e-01 -2.8289e-01
Virtual:
2.4902e-02 2.4902e-02 9.8705e-02 9.8705e-02 7.2829e-01
7.2829e-01 7.4939e-01 7.4939e-01 8.9189e-01 8.9189e-01
9.4246e-01 9.4246e-01 1.0499e+00 1.0499e+00 1.3587e+00
1.3587e+00 1.7578e+00 1.7578e+00 1.7729e+00 1.7729e+00
1.7987e+00 1.7987e+00 2.1415e+00 2.1415e+00 2.3604e+00
2.3604e+00
--------------------------------------------------------------------------------As you can see, we get a ground state energy of -76.4325580334 Hartrees. Additionally, the orbital section only has an Occupied and Virtual Section. This is due to the fact that 2-component methods do not have separate \alpha and \beta manifolds. Every orbital is a 2-component spinor.
Difference between GB3LYP
Since we have the X2C-B3LYP ground state energy. Let's find out how different it is from the non-relativistic GB3LYP ground state to understand the magnitude of the relativistic effects for water. To compute the GB3LYP energy, the only thing that needs to be changed in the above input file is the keyword reference = GB3LYP.
The resulting GB3LYP energy is then -76.3837898356 Hartrees. Taking the difference with the X2C solution, we see that the relativistic effects lower the total ground state energy by 0.0487681978 Hartrees, or roughly 1.3 eV. This discrepancy only gets larger for molecular systems with heavier elements.
References
-
Iliaš, M., & Saue, T. (2007). An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation. The Journal of Chemical Physics, 126(6), 064102.
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