Formaldehyde: Spin-Orbit Coupling¶
Spin-orbit coupling (SOC) mixes singlet and triplet states and controls the rate of intersystem crossing and phosphorescence. This example computes the singlet and triplet states of formaldehyde with MRSF-TDDFT and the SOC matrix elements between them, including the one-electron spin-orbit term and a mean-field treatment of the two-electron term. The result shows El-Sayed's rule directly.
Copy and run¶
mrsf(nstate=3)/bhhlyp/6-31g* soc
geom="""
C 0.000000 0.000000 -0.526342
O 0.000000 0.000000 0.676110
H 0.000000 0.937620 -1.119884
H 0.000000 -0.937620 -1.119884
"""
openqp formaldehyde-soc.oqp
mrsf(nstate=3) with soc computes three singlets (S0, S1, S2) and three
triplets (T0, T1, T2), then the SOC between every singlet–triplet and
triplet–triplet pair. The run takes about 4 seconds.
What you get¶
The SOC_MRSF section first lists every matrix element in cm⁻¹, split into
one-electron (1e) and two-electron (2e) parts. The column Absolute is
|1e + 2e| for that spin component. The largest elements are:
State_i State_j Re(1e) Im(1e) Re(2e) Im(2e) Absolute
S0 / T0(Ms= 0) 0.0000 -90.0740 0.0000 35.2698 54.804170567800
S1 / T0(Ms= 0) 0.0000 0.0000 0.0000 -0.0000 0.000000000003
S1 / T1(Ms= 0) 0.0000 -82.0876 0.0000 31.4124 50.675189621085
It then diagonalizes the full singlet–triplet Hamiltonian and lists the spin-orbit states with their composition:
State cm-1 eV Composition
1 -0.15 -0.000019 S0 (100.0%)
2 30971.21 3.839941 T0(+1) ( 50.0%) T0(-1) ( 50.0%)
3 30971.21 3.839942 T0(-1) ( 50.0%) T0(+1) ( 50.0%)
4 30971.45 3.839971 T0( 0) (100.0%)
5 32960.08 4.086530 S1 (100.0%)
6 47753.41 5.920669 T1(-1) ( 50.0%) T1(+1) ( 50.0%)
7 47753.43 5.920672 T1(+1) ( 50.0%) T1(-1) ( 50.0%)
8 47753.45 5.920674 T1( 0) (100.0%)
| State | Character | Energy above S0 |
|---|---|---|
| T0 | ³nπ* | 3.840 eV |
| S1 | ¹nπ* | 4.087 eV |
| T1 | ³ππ* | 5.921 eV |
| Coupling | Reference value |
|---|---|
| ⟨S0 | H_SO | T0, Ms = 0⟩ | 54.80 cm⁻¹ |
| ⟨S1 | H_SO | T0⟩ | 0 (all components) |
| ⟨S1 | H_SO | T1, Ms = 0⟩ | 50.68 cm⁻¹ |
Reading the result¶
- El-Sayed's rule. S1 and T0 are both nπ* states with the same orbital occupation, and their coupling is zero. S1 couples strongly to T1 (ππ*), where the orbital type changes. Intersystem crossing from S1 therefore needs the ππ* triplet or vibrational distortion of the molecule.
- S0–T0 coupling (54.8 cm⁻¹) controls phosphorescence from the ³nπ* state back to the ground state.
- The two-electron term matters. It has the opposite sign to the one-electron term and reduces the coupling by about 40%; a one-electron-only calculation would overestimate SOC.
- The spin-orbit splitting of each triplet is below 1 cm⁻¹ in this light molecule, so the spin-orbit states keep almost pure singlet or triplet character.
Try next¶
- Request different numbers of singlets and triplets with
soc(ns=3,nt=5)and a route withoutnstate:mrsf/bhhlyp/6-31g* soc(ns=3,nt=5). - Pyramidalize the molecule by moving both hydrogens a little along x, and compare the S1–T0 coupling with the planar value of zero.
See also: Spin-Orbit Coupling, MRSF-TDDFT.