Summary

Structure DFTB DFTB DFTB DFTB DFTB3 CCSD(T)

(ons) (3ord) (ons + 3ord) γ γ^h calc fit

4444-a 12.01 68.26 16.58 67.03 33.99 13.63 14.92 -171.06

4444-b 10.97 68.20 15.67 66.90 33.45 12.99 14.26 -170.52

antiboat 9.50 67.82 12.22 68.14 35.80 14.22 15.86 -170.55

boat-a 9.91 68.73 12.61 69.04 36.48 14.86 16.48 -170.80

boat-b 10.10 68.67 12.77 68.98 36.46 14.93 16.55 -170.64

MUDBE 10.5 68.3 14.0 68.0 35.2 14.1 15.6

MUD_RE 1.2 0.5 2.4 1.2 1.7 1.0 1.2

CE 0.56 1.83 0.95 2.02 1.31 0.61 0.70

Table 3.3: Comparison of binding energies of five water 16-mers as obtained with the DFTB method at different levels of approximation and CCSD(T). For all DFTB variants, the reported values are the deviations from the CCSD(T) results. The five water configurations are depicted in Fig. 3.2. At the bottom of the table, we provide the mean unsigned errors in the five binding energies (MUD_BE) as well as the mean unsigned errors in the ten relative energies of every combination pair (MUD_RE). Truhlar’s characteristic error (CE) is also given for every DFTB variant. All energies are expressed in kcal/mol.

As a final check, we computed the binding energies for the two water heptadecamers studied in Ref. [203] using DFTB(ons) and DFTB3(cal). Both methods also perform fairly well in this case, with MUD_BE of 10.1 and 15.6 kcal/mol and relative energy errors of 0.26 and 0.27 kcal/mol, respectively. These results suggest that both the onsite-corrected DFTB and DFTB3 method may be reliably employed for the study of some properties of neutral bulk water at a little computational cost.

this property. If, in addition, a third order extension is combined with our correction, hydrogen bond energies are further improved. Within such a scheme, the transferability to treat different charge states in hydrogen bonded systems clearly surpasses that of DFTB3 whereas no empirical parameters are necessary. The description of larger neutral water clusters is also improved with respect to standard DFTB results when employing onsite corrections. In this case, however, the combination of third-order extensions with our refinement does not outperform the onsite-corrected DFTB. We additionally showed that DFTB3 also describe these systems accurately.

The onsite correction has again the advantage of requiring no fitted parameters.

Chapter 4

THE TIME DEPENDENT DFTB METHOD

Although Casidas’ TD-DFT has demonstrated to be highly efficient, there are still many appli-cations in photochemistry and nanophysics out of the scope of the method. Quantum molecular dynamics in the excited state, for example, require the evaluation of energies and forces at a large number of points along the trajectory. Also, the investigation of extended nanostructures with intrinsic defects or surface modifications can not be reliably performed with small models.

These kind of problems might be addressed with an approximate TD-DFT formalism. Such a scheme is the time-dependent density functional based tight-binding method (TD-DFTB).

The development of TD-DFTB dates back to the year 2001 when Niehaus and coworkers, prompted by the good performance of ground-state DFTB, decided to extend the method to account for excited state properties [91]. The idea behind was simple. In density functional response theory, aside from the KS eigenpairs one needs to evaluate the coupling matrix. This matrix can be expressed as a sum of multicenter integrals by expanding the KS orbitals into the AO basis set. Next, it is straightforward to apply the same techniques for the evaluation of multicenter integrals that were earlier employed for second-order DFTB, namely, the Mulliken approximation and the monopole truncation of a MO multipole expansion. These approaches grant TD-DFTB users with the same advantages they savor with DFTB, that is, avoiding expensive integrations on the fly in favor of using pre-calculated two-center parameters. This results in a numerically efficient tool giving fast, yet fairly accurate results for demanding calculations.

The method was originally referred to as the γ approximation. The introduced parameters are very similar to the so-called γ and W constants. The only difference rests on that the new parameters are based on the actual electron density whereas γ and W are computed using neutral spin-unpolarized atoms. As the KS density is not known a priori, the new set of parameters would need to be determined during the calculation and the main asset of TD-DFTB would be lost. It was claimed, however, that the dependence of these quantities on the atomic net charges is negligible at least for systems with small charge transfer. Accordingly, TD-DFTB employs the known γ and W constants.

After the original implementation [91], TD-DFTB was extended in a number of different di-rections. The derivation of analytical excited state gradients [208] allows for the calculation of adiabatic transition energies and excited states geometries. Also, a real time propagation

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of KS orbitals using order-N algorithms has been derived [209]. Other extensions include non-adiabatic molecular dynamics simulations in the Ehrenfest [210] or surface hopping [211, 212]

approach as well as a TD-DFTB approach for open boundary conditions in the field of quantum transport [213]. A recent detailed review on the advantages and limitations of the method has been provided by Niehaus et al. [214].

In this chapter, we will follow a different strategy for the derivation of TD-DFTB. The coupling matrix will be obtained directly from the ground state theory. It will be shown that no addi-tional approximation or neglect is required for a linear response treatment within DFTB. The following derivation will also clearly justify the employment of the ground-state DFTB param-eters in TD-DFTB. Additionally, the method will be extended to account for spin-polarized systems and fractional occupation of the KS orbitals. It will be also shown that onsite correc-tions in TD-DFTB lead to important improvements over the traditional formalism.

4.1 Spin-unrestricted TD-DFTB

In section 3.4 we introduced a new formulation of DFTB, where the energy functional and KS Hamiltonian are expressed in terms of the KS density matrix fluctuations. This formalism is especially suitable for the derivation of a TD-DFTB scheme as the derivative of the KS Hamiltonian with respect to the density matrix elements (namely, the coupling matrix) is obtained straightforwardly. Thus, the coupling matrix is in this case a subblock¹ of the matrix K¯ defined in Eq. (3.40):

K_iaσ,jbτ := ∂H_iaσ

∂P_jb^τ = ¯K_iaσ,jbτ. (4.1)

This quantity depends on the γ and W parameters, as well as on q_Al^stσ, introduced earlier in Eq. (3.39). In this case, for which s = i and t = a, the quantities, q^iaσ_Al , are called Mulliken transition charges [91] and the matrix ˜P^iaσ, defined in Eq. (3.38), represents the dual KS transition density for an excitation from orbital i to a. It should be then clear that in TD-DFTB no further approximations are needed, other than those already introduced in TD-DFTB.

From Eq. (3.38), it should be noted that the dual transition density matrix is invariant with respect to the permutation of the indices i and a, that is, ˜P^iaσ = ˜P^aiσ. This implies that in TD-DFTB the coupling matrix is symmetric. This is, indeed, an expected property as typical DFTB is derived as an approximation to local or semi-local DFT². Due to this symmetry, the expression for the response matrix elements can be simplified as in Eq. (2.43). Note that the coupling matrix, Eq. (4.1), was derived from a spin-unrestricted formalism which, additionally, allows for occupancies such that 0 ≤ n_sσ ≤ 1, and hence, this generality is automatically transferred to the time-depended scheme.

It is worth formulating the method for closed shell systems as a particular case, in order to make contact with the original derivation of TD-DFTB . In this special case the Mulliken transition charges have the property q_Al^ia↑ =q_Al^ia↓ =q_Al^ia. If, in addition, the dependence of theγ-functional

1Recall that the coupling matrix is defined only for those elementsKiaσ,jbτsuch thatniσ > naσandnjτ > nbτ

whereas ¯Kis a more general matrix with no restriction or constraint on the orbitals.

2See Ref. [215] for an extension of DFTB to general hybrid functionals.

and W constants on the angular momentum is neglected, the coupling matrix simplifies to K_iaσ,jbτ =X

AB

q^ia_A(γ_AB+δ_σδ_τδ_ABW_A)q^jb_B, (4.2) with q_A^ia =P

lq_Al^ia. This expression is in full agreement³ with that derived previously for spin-unpolarized densities [91].

The singlet and triplet coupling submatrices are then expressed as K_ia,jb^S =Kia↑,jb↑+Kia↑,jb↓ =X

AB

q_A^iaq_B^jbγ_AB K_ia,jb^T =Kia↑,jb↑−Kia↑,jb↓ =X

A

q^ia_Aq_A^jbW_A (4.3)

Once the excitation vectors,F_iaσ^I , are obtained within TD-DFTB, the oscillator strengths can be computed from (2.47). The transition dipole matrix is then conveniently subject to a Mulliken approximation,

d_iaσ ≈X

A

R_Aq_A^iaσ, (4.4)

where R_A is the position of atom A.