It turns out that the E and Z isomers have different colors - one is
yellow and the other red - and the authors attribute this to a change in
the lower energy band, assigned as a n−π∗ transition. Let's
see how we could have predicted these results from first principles.
TD-DFT
瞬态密度泛函理论(time-dependent density functional theory (TD-DFT)
)是最常用用于预测紫外/可见光谱的方法，简单且廉价2.
ORCA使用，B3LYP 三重态基组
Here we also used the CPCM as a solvation model for both the ground
and the excited states to simulate the hexane used in the experiment,
which for the excited state case means the LR-CPCM model by default.
After the regular SCF, the TD-DFT header is printed:
Input orbitals are from ... AZOZ_TDDFT_HEX.gbw CI-vector output ... AZOZ_TDDFT_HEX.cis Tamm-Dancoff approximation ... operative CIS-Integral strategy ... AO-integrals Integral handling ... AO integral Direct Max. core memory used ... 4000 MB Reference state ... RHF Generation of triplets ... off Follow IRoot ... off Number of operators ... 1 Orbital ranges used for CIS calculation: Operator 0: Orbitals 16... 54 to 55...567 XAS localization array: Operator 0: Orbitals -1... -1
Dimension of the eigenvalue problem ... 20007 Number of roots to be determined ... 30 Maximum size of the expansion space ... 300 Maximum number of iterations ... 100 Convergence tolerance for the residual ... 1.000e-06 Convergence tolerance for the energies ... 1.000e-06 Orthogonality tolerance ... 1.000e-14 Level Shift ... 0.000e+00 Constructing the preconditioner ... o.k. Building the initial guess ... o.k. Number of trial vectors determined ... 300
****Iteration 0****
Memory handling for direct AO based CIS: Memory per vector needed ... 14 MB Memory needed ... 1329 MB Memory available ... 3000 MB Number of vectors per batch ... 203 Number of batches ... 1 Time for densities: 0.503 Time for RI-J (Direct): 3.903 Time for K (COSX): 45.512 Time for XC-Integration: 21.118 Time for LR-CPCM terms: 9.571 Time for Sigma-Completion: 0.393 Size of expansion space: 90 Lowest Energy : 0.092988177038 Maximum Energy change : 0.259251849588 (vector 29) Maximum residual norm : 0.016689588426 [...]
Note that the RI and COSX approximations are also used to compute the
necessary integrals for the TD-DFT solution. The error here is usually
even smaller than in the SCF, and the calculation runs
much faster (by a factor of about 10), so it is highly
recommended to use these.
Printing of the excited
states
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
# 输出收敛态 ------------------------------------ TD-DFT/TDA EXCITED STATES (SINGLETS) ------------------------------------
the weight of the individual excitations are printed if larger than 1.0e-02
STATE 1: E= 0.089757 au 2.442 eV 19699.5 cm**-1 <S**2> = 0.000000 50a -> 55a : 0.039008 (c= -0.19750352) 54a -> 55a : 0.942532 (c= 0.97084090)
STATE 2: E= 0.138303 au 3.763 eV 30354.1 cm**-1 <S**2> = 0.000000 53a -> 55a : 0.938086 (c= 0.96854831) 54a -> 58a : 0.027177 (c= -0.16485521) [...]
It is important to have in mind that these energies actually are the
vertical energy differences, or the energy of that state in the
ground state geometry.
Below the energies, the contributions of each single excitation are
printed, first with its relative contribution (these will be discussed
later), followed by the eigenvector value.
The states and their respective vertical transition energies are
printed again, now together with the oscillator strength and transition
dipole moments. Strictly speaking, this is yet not a "spectrum" like the
experimental one because we only have transition lines, not
bands.
Plotting the calculated
spectrum
What we can do to approximate the experimental spectrum then is, e.g,
convolute these lines with a Gaussian function to turn them into bands.
This can be done easily by using Avogadro: open the output file, then go
into `"Extensions"and then "Spectra..."to open
the "Spectra Visualization" window:
img
There you can click on "Advanced" and then
"Absorption Settings" to set the details such as the line
width and even "Export Spectra Data" to replot this graphic
using a different software:
Back to the example
Here is a plot of the predicted spectrum for both the E and Z isomers
obtained from the calculation above, using B3LYP and the DEF2-TZVP
basis:
The two low energy peaks at about 507 and 417 nm for the E and Z
isomers match quite well the experimental results of 490 and 404 nm. If
one converts this wavelength to color and takes its complementary, the
correspondence is fairly good:
It is not that common that these results agree so well to the
experiment, and here it can be somewhat accidental. It is much more
prevalent that the predicted spectra is "shifted" by ±0.1 to ±0.5 eV and
it is costumary in the literature to fix that, if there is any
reference, to correct systematic errors.
------------------------------------------ NATURAL TRANSITION ORBITALS FOR STATE 13 ------------------------------------------
Making the (pseudo)densities ... done Solving eigenvalue problem for the occupied space ... done Solving eigenvalue problem for the virtual space ... done Natural Transition Orbitals were saved in AZOE_TDDFT_HEX.s13.nto Threshold for printing occupation numbers 0.001000
the S13 is now described in terms of only two main transitions. These
NTOs are saved in a file named basename.s13.nto, and can be printed
using the orca_plot tool.
../_images/S13.png
which are clearly π−π∗ transitions on the aromatic rings, and S13S13
can then be assigned as such
A higher level method
- STEOM-DLPNO-CCSD
高等级计算方法 STEOM-DLPNO-CCSD
As shown above, TD-DFT can be successful in many cases when trying to
predict excited states, however one can not know a
priori if a given functional will work or not and different
functionals are bound to give different results.
In order to avoid this trial-and-error between functionals, an ab
initio unparametrized method is sometimes a better choice. In ORCA,
many of these methods are available, but for the sake of simplicity we
will show here one that has a very good accuracy, still at an affordable
computational cost: the Similarity Transformed Equation of Motion CCSD
(STEOM-CCSD).
The STEOM-CCSD is a method that also includes Correlation
energy into the calculation of the excited states, thus increasing
the quality of the prediction [Iszak2019a].
It is in itself a heavy method and very computational demanding, but the
ORCA team has recently developed The
DLPNO scheme version of it (DLPNO-STEOM), that presents a much
better scaling and it is not much costlier than TD-DFT [Iszak2019b].
In order to use that, just optimize your geometry somehow and
run:
As in other DLPNO or correlated methods using the RI, the /C basis is
necessary for the calculation. We also added the CPCM correction for
both the ground and excited states, here the latter is invoked by
setting DOSOLV TRUE under %MDCI, which is the block that controls these
calcualtions.
The MAXITER 100 defines the maximum number of iterations during the
many steps of STEOM, and sometimes it is necessary to increase it to
achieve convergence.
The output printed is much larger than that of TD-DFT, for there are
many more steps necessary to conclude the overall computation. In
summary what is done is:
First a regular DLPNO-CCSD calculation is performed.
That is followed by a simple CIS calcuation, that is necessary to
reduce the active space.
Then both a DLPNO-IP-EOM and a DLPNO-EA-EOM are made.
EOM type ... STEOM Multiplicity ... singlet Solver ... Davidson Convergence check ... for each root separately Convergence threshold ... 1.00E-05 Root homing ... on Preconditioning update ... CIS Reduced space size (times number of roots) ... 40 Number of roots in the CIS initial guess ... 600 Number of roots to be optimized ... 15 Number of amplitudes to be optimized ... 20007 [...]
The equations are solved rootwise by default, and after everything
converges, the results are printed:
Similarly to TD-DFT, the square of the amplitudes is the percentage
of that excitation in the final state. The solvent shift is done, some
other integrals are calculated and the "spectrum" is printed:
Here we added the gray vertical lines to guide the eye to the first
three peaks of the spectrum. As you can see, although TD-DFT works well
to predict the first, the higher energy ones are largely displaced.
That can happen when these states are significantly different, and
the associated error is not comparable. One advantages of the STEOM is
that there is a more balanced treatment of different excited states and
such situations and avoided.
The calculated results are now well in line with the experiment
through the whole spectrum, except for a small deviation on the
intensities. We can arrange these results in a table to facilitate
visualization: below there is comparison of the energy error of the
first three peaks measured for the E isomer. The TD-DFT can be as high
as 1 eV!
Peak
TD-DFT
STEOM
Exp.
1
4.23
5.06
5.14
2
3.83
4.42
4.43
3
2.46
2.45
2.53
Even if the predicted spectrum matches the experiment, the
calculation is still not reproducing exactly what happens
during light absorption. The full experimental spectra must also include
vibrational resolution, and maybe even vibronic coupling. That can be
included with the ORCA_ESD module.
