4.9. Conical Intersections¶
The minima in the conical intersection seam-space between two states (named here I and J) can be found by using regular geometry optimization algorithms, except that the gradient to be optimized is [563]:
where \(\mathbf{g'}_{diff} = 2(E_I-E_J)(\partial E_I / \partial q - \partial E_J / \partial q)\) is parallel to the gradient difference vector; \(\mathbf{g}_{mean}\) is the gradient mean and \(\mathbf{P}\) is a projection matrix that projects out the gradient difference (\(\mathbf{x}\)) and non-adiabatic coupling (\(\mathbf{y}\)) direction components:
Now we have three approaches to solve this problem in ORCA, that will be explained next.
4.9.1. Gradient Projection¶
This is exactly what has been described above, and will be chosen as default whenever NACMEs between I and J are available. It is in principle the fastest and most accurate method. It can be invoked by setting:
%CONICAL
METHOD GRADIENT_PROJECTION #or simply GP
END
OBS.: Turning on the ETF (see Sec. Section 5.6.18.1) can improve the optimization when using the full Gradient Projection method.
4.9.2. Gradient Projection without NACME¶
It is an approximation to the method above, that one gets by completely neglecting the NACMEs. It is essentially equivalent to finding a surface crossing point, and will not necessarily find minima inside the CI seam-space, although the final \(\Delta E_{IJ}\) should be zero.
%CONICAL
METHOD GP_NONACME
END
4.9.3. Updated Branching Plane¶
Here the idea is to start from a guess NACME, which is any unit vector perpendicular to \(\mathbf{x}\), and do a progressive update on it, similar to the BFGS update on the Hessian [563]. The “Branching Plane” defined by \(\mathbf{x}\) and \(\mathbf{y}\) then becomes iteratively more accurate until covergence is achieved. It has been shown to be quite accurate and is the default whenever NACMEs are not available. Can be used with:
%CONICAL
METHOD UBP
END
Finally, the \(\Delta E_{IJ}\) energy threshold for the optimization can be altered with:
%CONICAL
ETOL 1e-4 #default
END
As a further example for such an analysis we show the conical intersection optimization of the ground and first excited state of singlet ethylene.
Note
This is currently only available using TD-DFT, will be expanded in future versions. More details about the specific options on Excited States via RPA, CIS, TD-DFT and SF-TDA.
Note
Even though locating the CI of a TD-DFT excited state and the reference state is supported, it is not the recommended way of finding the ground state-excited state CI, because such CIs are not described properly by TD-DFT (in particular, TD-DFT even predicts the wrong dimensionality for the intersection space). Instead, it is advised to use SF-TD-DFT for this purpose, e.g. use the \(T_1\) state as the reference state, and calculate both the \(S_0\) and \(S_1\) states as excited states. (vide infra)
!B3LYP DEF2-SVP CI-OPT
%TDDFT IROOT 1 END
* xyz 0 1
C 0.595560237 -0.010483480 -0.000284187
C -0.831313750 0.167231832 0.001482505
H -1.381857976 0.227877089 0.963419721
H 1.265119434 0.874806815 0.006897459
H -1.382258208 0.243775568 -0.959090898
H 1.027489724 -1.032962768 -0.008829646
*
Tip
You can often use a structure between the optimized structures of both states for your CI-optimization. If this doesn’t work, you might try a scan to get a better initial guess.
The results of the CI-optimization are given in the following output. The energy difference between the ground and excited state is printed as E diff. (CI), being reasonably close for a conical intersection. For a description of the calculation of the non-adiabatic couplings at this geometry, see section Section 5.6.18.2.
.--------------------.
----------------------|Geometry convergence|-------------------------
Item value Tolerance Converged
---------------------------------------------------------------------
Energy change 0.0000164283 0.0000050000 NO
E diff. (CI) 0.0000025162 0.0001000000 YES
RMS gradient 0.0000068173 0.0001000000 YES
MAX gradient 0.0000136891 0.0003000000 YES
RMS step 0.0000358228 0.0020000000 YES
MAX step 0.0000821130 0.0040000000 YES
........................................................
Max(Bonds) 0.0000 Max(Angles) 0.00
Max(Dihed) 0.00 Max(Improp) 0.00
---------------------------------------------------------------------
Everything but the energy has converged. However, the energy
appears to be close enough to convergence to make sure that the
final evaluation at the new geometry represents the equilibrium energy.
Convergence will therefore be signaled now
***********************HURRAY********************
*** THE OPTIMIZATION HAS CONVERGED ***
*************************************************
---------------------------------------------------------------------------
Redundant Internal Coordinates
--- Optimized Parameters ---
(Angstroem and degrees)
Definition OldVal dE/dq Step FinalVal
----------------------------------------------------------------------------
1. B(C 1,C 0) 1.3254 -0.000005 -0.0000 1.3254
2. B(H 2,C 1) 1.1270 0.000004 -0.0000 1.1270
3. B(H 3,C 0) 1.1271 -0.000002 0.0000 1.1271
4. B(H 4,C 1) 1.1271 0.000000 -0.0000 1.1271
5. B(H 5,C 0) 1.1271 -0.000002 0.0000 1.1271
6. A(H 3,C 0,H 5) 106.00 0.000001 -0.00 106.00
7. A(C 1,C 0,H 5) 126.97 -0.000013 0.00 126.97
8. A(C 1,C 0,H 3) 127.03 0.000011 -0.00 127.03
9. A(C 0,C 1,H 4) 127.03 0.000013 -0.00 127.03
10. A(H 2,C 1,H 4) 106.01 -0.000000 0.00 106.01
11. A(C 0,C 1,H 2) 127.04 0.000009 -0.00 127.04
12. D(H 2,C 1,C 0,H 5) 106.86 -0.000005 0.00 106.86
13. D(H 4,C 1,C 0,H 3) 106.76 -0.000005 0.00 106.76
14. D(H 4,C 1,C 0,H 5) -73.62 -0.000004 0.00 -73.61
15. D(H 2,C 1,C 0,H 3) -72.76 -0.000005 0.01 -72.76
----------------------------------------------------------------------------
*******************************************************
*** FINAL ENERGY EVALUATION AT THE STATIONARY POINT ***
*** (AFTER 12 CYCLES) ***
*******************************************************
4.9.4. CI Minima Between Excited States¶
In an analogous way, the conical intersection minima between two excited states can be requested by selection both an IROOT and a JROOT, shown below.
!B3LYP DEF2-SVP CI-OPT
%TDDFT IROOT 2
JROOT 1
#IROOTMULT TRIPLET would search in the triplet PES
#SF TRUE would search for the S0-S1 CI from a T1 reference, using SF-TD-DFT
# (but remember to set the multiplicity as 3 instead of 1)
END
* xyz 0 1
C 0.595560237 -0.010483480 -0.000284187
C -0.831313750 0.167231832 0.001482505
H -1.381857976 0.227877089 0.963419721
H 1.265119434 0.874806815 0.006897459
H -1.382258208 0.243775568 -0.959090898
H 1.027489724 -1.032962768 -0.008829646
*