PE-TDDFT calculations of excitation energies and solvent shifts

In this tutorial we investigate property calculations in solution through the polarizable embedding (PE) model ([Olsen2010], [Olsen2011]). The tutorial builds on the implementation reported in [Hedegaard2017] which uses PElib to calculate the environment contributions. This implementation allows calculation of electric properties up to linear response (this corresponds to electric properties up to second order), including an explicit environment. Examples include electric dipole moments and polarizabilities. In addition, the linear response module allows calculation of excitation energies and transition moments, including the effect of a surrounding environment through the PE model. Here we focus on TDDFT including a solvent. A few comments regarding calculations of environment effect on dipole moments and polarizabilities are given in the end of this tutorial.

TDDFT including an explicit environment through the PE model

The generalized eigenvalue problems that is solved when solving the response equations

\begin{equation*} \left(E^{[2]} -\hbar\lambda S^{[2]}\right)X_m=0 \end{equation*}

which is completely equivalent to what is done in a vacuum calculation. The main difference is in the electronic Hessian, \(E^{[2]}\) , which is modified by the presence of an environment (cf. Eqs. 36-38) in [Hedegaard2017]. The Hessian has the structure

\begin{equation*} E^{[2]} = \left(\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^* & \mathbf{A}^{*} \end{array} \right) , \end{equation*}

with matrix elements

\begin{align*} A_{ai;bj} = \langle 0 \vert [\hat{q}^{\dagger}_{ai},[\hat{q}_{bj},\hat{f}_0 + \hat{v}^{\mathrm{PE}}]] \vert 0 \rangle + \langle 0 \vert [\hat{q}^{\dagger}_{ai},\hat{v}^{\mathrm{ind}}]] \vert 0 \rangle \label{Hessian_2a} \\ B_{ai;bj} = \langle 0 \vert [\hat{q}_{ai},[\hat{q}_{bj},\hat{f}_0 + \hat{v}^{\mathrm{PE}}]] \vert 0 \rangle + \langle 0 \vert [\hat{q}_{ai},\hat{v}^{\mathrm{ind}}] \vert 0 \rangle . \end{align*}

The modification is manifested in the additional terms \(\hat{v}^{\mathrm{PE}}\) and \(\hat{v}^{\mathrm{ind}}\) , while \(\hat{f}_0\) is the usual Fock (or Kohn-Sham) operator. A brief, physical explaination of the the two additional terms is given below

  • \(\hat{v}^{\mathrm{PE}}=\hat{v}^{\mathrm{es}}+\hat{v}^{\mathrm{ind}}\) accounts for interactions between multipoles and ground-state density \(\hat{v}^{\mathrm{es}}\) as well as mutual polarization between the ground-state density and the environment \(\hat{v}^{\mathrm{ind}}\) .
  • \(\hat{v}^{\mathrm{ind}}\) accounts for the differential interaction between the ground- and excited states with a polarizable environment.

A derivation and more thorough explanation of the \(E^{[2]}\) term can be found in [Hedegaard2017] (cf. Eqs. 36-38).

To demonstrate how to calculate the effect of the environment with PE, we use the same system as in the PE-HF tutorial. The employed h2o.mol file and potential 3_h2o.pot can be found in that tutorial. The input pe_response.inp is similar to a regular TDDFT calculation but with an additional .PEQM keyword under the **HAMILTONIAN section:

**DIRAC
.TITLE
H2O
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
.DFT
B3LYP
.PEQM        ! activate polarizable embedding
**INTEGRALS
*TWOINT
.SCREEN
1.0D-12
*READIN
.UNCONT
**WAVE FUNCTIONS
.SCF
*SCF
.CLOSED SHELL
10
**PROPERTIES
*EXCITATION ENERGIES
.EXCITA
1 5
.INTENS
0
**END OF

As for the case of PE-HF, the specification of .PEQM will thus make the program calculate the PE-specific terms; in case of linear response that is the additional \(\hat{v}^{\mathrm{PE}}\) and \(\hat{v}^{\mathrm{ind}}\) terms. Note that use of the .GSPOL keyword under the *PEQM flag will result in a calculation where only the term involving \(\hat{v}^{\text{PE}}\) is included (physically, this corresponds to a frozen environment during the excitation process). Now we have all ingredients for starting the calculation:

pam --inp=pe_response.inp --mol=h2o.mol --pot=3_h2o.pot

The summary will look identical to a vacuum calculation:

* Isotropic DL-DL non-zero oscillator strengths (f)
===================================================
DL = dipole length
Rate = Dipole radiation rate (s-1)
Lifetime = corresponding radiation lifetime (s)

Level  Frequency (eV)    f           Rate         Lifetime    Symmetry
------------------------------------------------------------------------
    1      7.00316    0.000001    1.26019E+03    7.93528E-04
    2      7.00317    0.000000    3.70356E+02    2.70010E-03
    3      7.00317    0.000002    3.30199E+03    3.02848E-04
    4      7.30170    0.049543    1.14615E+08    8.72484E-09
    5      8.53008    0.000000    5.32729E+01    1.87713E-02
------------------------------------------------------------------------
Sum of oscillator strengths:      0.04955

Note that the first three excitations have very low oscillator strength. These excitations corresponds to triplet excitations, which are automatically included in a relativistic framework. The fourth excitation is the observed one. We can calculate the electrostatic part of the solvent shift (i.e. not including the solvent effect on the geometry), by calculating the corresponding excitation energy without PE and substracting the result from the PE result. We collect results with and without PE (and the obtained solvent shift) in the table below.

QM method (potential method) Potential PE vac. shift
B3LYP (B3LYP/6-31+G*) 3 H2O 7.30 6.69 0.61
Experimental   8.2 7.4 0.8

The experimental value is taken from discussion given in [Christiansen2000]. We note that

  • The shift is a bit below the experimental one
  • The excitation energies are somewhat underestimated (mainly due to the B3LYP functional).

To improve the model, we can either (i) expand the environment, (ii) describe the environment more accurately or (iii) describe the QM system more accurately (but currently DIRAC only includes PE with HF and DFT). Finally, we might also (iv) expand the QM system. Performing these calculations will be not be part of this tutorial. A table with selected results from larger environments, potentials obtained with a more accurate basis set (aug-cc-pVTZ) and a more accurate QM method are compiled below. All these calculations are non-relativistic calculations carried out with the Dalton program and taken from [Hedegaard2016]. The “full CI” calculations are obtained with the density matrix renormalization group (DMRG).

QM method (potential method) Potential PE vac. shift
Non. rel. B3LYP (B3LYP/6-31+G*) 127 H2O 7.65 6.69 0.96
Non. rel. full CI (B3LYP/6-31+G*) 3 H2O 7.90 7.46 0.44
Non. rel. full CI (B3LYP/6-31+G*) 127 H2O 8.17 7.46 0.71
Non. rel. B3LYP (B3LYP/aug-cc-pVTZ) 3 H2O 7.42 6.69 0.73
Non. rel. B3LYP (B3LYP/aug-cc-pVTZ) 127 H2O 7.89 6.69 1.20
Non. rel. full CI (B3LYP/aug-cc-pVTZ) 3 H2O 7.95 7.46 0.49
Non. rel. full CI (B3LYP/aug-cc-pVTZ) 127 H2O 8.36 7.46 0.90

Electric dipole moments and polarizabilities

It is also possible to calculate simple expectation values or other linear response properties with the PElib module. The work-flow is straightforward, and we only need to add the .PEQM keyword as done for the TDDFT calculation above. The input related to the property is identical to a vacuum calculation. Examples of simple expectation values and linear response properties that are currently implemented with PE is the (electric) dipole moment and static or frequency-dependent polarizabilities. Unlike in a non-relativistic framework, the relativistic framework allow many magnetic properties to be obtained simply as expectation values (e.g. properties related to electron paramagnetic spectroscopy). All PE contributions to these properties will be automatically included from the wave-function optimization and will follow in a future release of an EPR module. Note also that linear response properties that require London orbitals are also not currently possible with PE. An example is given below for a dipole moment calculation input file

**DIRAC
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
.PEQM
**WAVE FUNCTION
.SCF
**PROPERTIES
.DIPOLE
*END OF

Things to note

  • Calculations with transformed two-component Hamiltonians are generally possibly by supplying the program with the requested model under **HAMILTONIAN. In such a calculation, \(\hat{v}^{\mathrm{PE}}\) is not transformed