Bath operators in the single-particle KS framework and related features 95

The choice of the bath operator is at the heart of the applicability of the open quantum system TDDFT scheme. On the one hand, the bath operator models the underlying physics of the bath mechanism, thus needs to be motivated by physical processes. On the other hand, it is important for the theoretical foundation of the open quantum system TDDFT framework and the existence of a single-particle scheme. Here, I introduce four heuristically motivated bath operators. All four bath operators are designed for the single-particle framework of KS TDDFT. They involve projection operators and induce relaxation of the entire excited system back to its ground state via projection to the occupied ground-state KS orbitals. The rationale behind these operators and their implementation is explained in the following.

The central idea behind the rst bath operator is to interpret the dierences between eigenvalues of occupied and unoccupied KS orbitals as excitation energies of the system. In this sense, one obtains, e.g., an excited state by replacing one of the occupied orbitals of the GS system by one of the higher lying unoccupied KS orbitals. Then, relaxation to the ground state may be performed if one detects the overlap of the time evolution of each orbital of this system with the space of all (or at least the most important) unoccupied orbitals and projects back to the ground state. To model this behavior with single-particle operators, the rst bath operator s⁽¹⁾_i uses a large basis of M occupied and unoccupied KS orbitals, computes the overlap of all TD KS orbitals to these basis functions, and projects onto the corresponding GS orbital depending on the magnitude of the overlap [PDV08, ADV11]. This bath operator reads

s⁽¹⁾_i =√γ XM j=i+1

|ϕ_i(t₀)ihϕ_j(t₀)|, (D.41) where I use all KS eigenstates with an eigenvalue that is energetically above the eigenvalue of the reference orbitalϕi(t₀)and project back to the latter orbital. The factor√γincludes the

decay rateγ corresponding to the decay time τ = 1/γ. Although this type of bath operator may be used to model interesting physics, to the best of my knowledge there is no explicit proof available that guarantees for its validity in the stochastic TDDFT framework of open quantum system theory. Therefore, I investigated alternative types of bath operators for the work that is presented in Sec. 5.3 and used s⁽¹⁾_i only for test calculations and implementing the algorithms.

The second class of bath operators induces relaxation of the excited system back to its ground state via projectors of the TD KS orbitals onto the corresponding GS orbitals. Such operators are motivated and introduced in Sec. 5.3.2. Here, I shortly outline a generalized version of this bath operator together with its implementation to the PARSEC code. The time constant of the decay process is given by the decay rateγ. One additional multiplicative factor renders this type of bath operator sensitive to variations of the multipole moments of the entire density or to such moments of particular subsystems. For reasons of a notation consistent with the PARSEC input that is explained in Sec. D.3.3, this bath operator is labeled by the superscript (4) and reads

s⁽⁴⁾_i =√γ|d_k(t)−d_k(t₀)|

D |ϕi(t₀)ihϕi(t)|, (D.42) whereγ is an eective decay rate. The operator is proportional to variations|dk(t)−dk(t0)| of the dipole momentd_k(t) =R

rn_k(r, t) d³rof the subsystem indicated by the indexk. Here, the indexk denotes a specic part of the grid using a particular partitioning scheme of the grid (see Sec. D.1 for further details about grid partition ideas). The density nk(r, t) is the density that corresponds only to subsystemk. Thus, this operator causes deexcitation of the entire system, but is sensitive to local changes of the dipole moment from the corresponding GS values during time propagation. Ddenotes a normalization factor of the dipole moment variations. In this approach, the time constant related to the rate γ of the dissipative mechanism is a free parameter, whereas D needs to be chosen reasonably as discussed in detail in Sec. 5.3.2. An attempt towards the theoretical justication of the use of this bath operator modulo the dipole-dependent factor in the framework of stochastic TDDFT using single-particle KS equations is outlined in Sec. D.3.1.

Another approach to dene bath operators involves besides to the KS orbitals also lo-calized orbitals {ϕ^loc_κ (t₀)}, where localization is performed according to the Foster-Boys criterion. The rationale behind this idea is to nd bath operators that act in a restricted part of the grid. To this end, they measure the spatial localization in a specic region of the grid by calculating the overlap of the K localized orbitals ϕ^loc_κ (t₀) and the KS orbital onto which the operator acts. Here, the localized orbitals are dedicated and selected to rep-resent orbitals that correspond to subsystems localized in particular parts of the entire grid.

Therefore, instead of using the KS orbitals as ins⁽⁴⁾_i , the bath operators⁽²⁾_i uses Foster-Boys orbitals in the projector that are computed from localization of the occupied ground-state KS orbitals and projects onto the corresponding ground-state KS orbital, i.e.,

s⁽²⁾_i = XK κ=1

√γ|d_k(t)−d_k(t₀)|

D |ϕ_i(t₀)ihϕ^loc_κ (t₀)|. (D.43) The dipole-moment-sensitive factor guarantees for the dependence of the action of this bath operator on the dipole moment. Alternatively, bath operator s⁽³⁾_i resembles the previous

D.3. STOCHASTIC TIME-DEPENDENT DENSITY FUNCTIONAL THEORY 97 operator, but does not include the dipole-moment-sensitive factor. Instead, it uses Foster-Boys orbitals computed not only from the occupied GS orbitals, but also takes unoccupied KS orbitals into account. Among these, PARSEC selects all Z orbitals ϕ^loc_ζ (t0) that are localized in a predened part of the grid. The bath operator reads

s⁽³⁾_i = XZ ζ=1

√γ|ϕi(t₀)ihϕ^loc_ζ (t₀)|. (D.44)

Yet, to the best of my knowledge, the application of bath operators that rely on localized orbitals computed from ground-state KS orbitals in the context of stochastic TDDFT is not based on a solid theoretical proof. The rst tests of these operators revealed diculties that might be related to the interplay of localized Foster-Boys and delocalized KS orbitals in single-particle projection operators. Nevertheless, using localized orbitals to design spatially localized bath operators might be a promising route for further bath-operator development based on projection operator formalisms. In particular, it might be interesting to investi-gate the application of localized orbitals computed from the TD KS orbitals during time propagation.

For bath operators that relax the excited system back to its ground state, some prac-tical advantages appear in the quantum-jump algorithm of Sec. 5.2.2: If no TD external perturbations occur after the initial excitation, the system remains in the ground state after each quantum jump. As a consequence, the time evolutions of all orbitals of the stochastic ensemble follow the same pattern: The KS system evolves deterministically until a quan-tum jump occurs. Then, back in the ground state, the system propagates trivially with TD phases according to the KS eigenvalues as acting with the bath operator on the ground state does not change the system. Therefore, the entire statistical ensemble can be generated from a single deterministic evolution of Eq. (5.16) alongside with the determination of the waiting-time distribution:

(i) calculate a suciently long deterministic evolution together with the norm decay of the auxiliary system

(ii) draw random numbers, nd the points of time of a large number of quantum jumps, and determine the waiting-time distribution

(iii) generate single trajectories of the ensemble where the time evolution up to the jump is determined by the deterministic evolution and GS values are used for the time after the jump

The physical quantities of interest follow from averaging over the observables calculated from the thus obtained statistical ensemble. Finally, note that I perform orthogonalization at the end of each time step as suggested in Ref. [ADV11] in case of bath operators s⁽¹⁾_i to s⁽³⁾_i , whereas orthogonality is preserved in bath operator (4).

D.3.3 PARSEC features and input parameters

The nal section of this appendix is dedicated to an explanation of the PARSEC input parameters related to the stochastic TDDFT (STDDFT) implementation. Additional general remarks concerning the PARSEC input and especially the input data types are compiled in

Sec. E.1. All STDDFT functionalities may be (de)activated by the boolean ag Use_stddft.

When STDDFT calculations are performed, the most important choice is the selection of the bath operator via the integer input parameter Bath_Type. The numbers that specify the dierent operators correspond to the numbers in parenthesis that I used in the superscripts of Eqs. (D.41) to (D.44) to label the dierent operators in the previous section. One needs to specify these numbers using the integer input parameter Bath_operators to choose among the bath operators. Yet, in the current version only a single type of bath operator can be used per PARSEC run. Either the decay rate or the decay time need to be set for this bath operator using the input parameters Decay_rate (double precision) or Decay_time (physical).

In the cases of bath operators (2)-(4) that involve either dipole moments of local subsystems or Foster-Boys localized orbitals, one needs to specify the grid partition to which the bath operator is supposed to be sensitive. This information may be provided via the input block Damping_centers, where PARSEC expects the index of the desired partition (see Sec. D.1) in the rst column and, in the second column, the normalization factorDcorresponding to the molecule that is located in the indicated partition. The determination of this normalization factor is explained in Sec. 5.3.2. If localized orbitals are involved, the localization threshold may be set via the parameter Localization_threshold (double precision).

A partition-selective excitation may be performed as explained in Sec. D.1.3. Alterna-tively, in case of bath operator (1), excitation may be performed within the KS system by replacing one of the occupied orbitals of the KS GS conguration by one of the energetically higher lying unoccupied orbitals. The present PARSEC implementation provides means to replace the HOMO orbital by the LUMO. This feature can be (de)activated by the boolean ag Switch_orb. A partition-selective observation (see Sec. D.1.3) is automatically active if partition-selective excitation or damping are used. Additionally, one may also request output of the time evolution of the projection of each TD KS orbital onto the GS KS or-bitals via the boolean input ag Overlap_output. This option is not active by default. More output can be generated during STDDFT calculations if one is interested also in details of the norm decaying auxiliary system. This output may be selected via the boolean ag Decaying_system_output but is inactive by default.

Last but not least, the determination of the quantum-jump times is based on random numbers. However, working with random numbers causes diculties during the implemen-tation phase of such a stochastic code as one likes to preform identical calculations to test the inuence of changes in the code. Apart from this, for some of the calculations, I prefer to set up the random numbers outside the PARSEC code, thus perform PARSEC calculations with preset random numbers. For such cases, I introduced the integer input parameter Ran-dom_seed, that can be used to specify the initialization and thus predetermine the outcome of the random number generator.

Appendix E

PARSEC miscellaneous

This appendix comprises a collection of PARSEC functionalities that I implemented during the work on this thesis and that do not belong to the topics covered in the previous appen-dices. Sections E.2 to E.4 contain PARSEC features related to the real-time propagation routines, whereas in Secs. E.5 and E.6 I present some implementations to the ground-state section of the code. For a more general introduction to the PARSEC input, see the PAR-SEC documentation and Ref. [KMT⁺06]. In the following section, I start with some general comments about the design of the PARSEC input script.

E.1 General comments on the PARSEC input

The PARSEC input involves dierent structures and data types that may be used for a case-specic denition of input variables¹. Therefore, in the following I briey introduce the data types that are relevant for the input options added during the work on this thesis. Control parameters that determine specic functionalities of the code are typically based on the data types integer (int), boolean (bool), or string. For typical physical and numerical parameters, the data types integer, double precision (dp), and physical are used. The dierence between double and physical is that in case of variables that are indicated to be physical, a physical unit corresponding to the specic input parameter may be added to the input value. If no physical unit is specied explicitly in the input le, PARSEC reads the parameter value nevertheless and assumes default units. The PARSEC input block structure (block) allows for the input of data arrays in terms of the lines and columns between the block-structure indicators. In such structures, all numbers are assumed to be of double precision value. If necessary, I indicate these data types during the explanations about the input parameters that are relevant for this work.

E.2 Determining the midpoint Hamiltonian during