Development of formalisms based on locally coupled open subsystems for calculations in molecular electronic structure and dynamics

Quantum embedding theories model a collection of interacting molecules as a set of subsystems, where each can be treated with a particular electronic structure method (wave function or density functional theory, for example); these theories can lead to computationally efficient and accurate algorithms. Motivated by challenges in the field, we previously described a formalism (that models two subsystems), which we call “locally coupled open subsystems” (LCOS), for the computation of ground-state energies, fractional electron-occupation numbers of the subsystems, and the size-consistent limit of subsystem dissociation. In this work we present the full (nonrelativistic) LCOS theory and the following extensions of our previous work: (i) the framework to study systems composed of multiple subsystems and a procedure to spin-adapt the auxiliary wave function that describes the partitioned system, so that its spin state matches that of the real system of interest; (ii) potential functionals and ideas to employ machine learning for the computation of ground-state densities and energies; (iii) formulation of two LCOS ground-state approaches where the fragments are assigned Kohn-Sham wave functions; and (iv) a time-dependent (TD) extension of these two ground-state formalisms in which the state of the subsystems evolves according to a unitary propagation; from this evolution we can extract TD electron populations of the fragments, for instance. We also discuss potential applications of the TD LCOS theory to linear photoabsorption and Raman spectroscopy. The developments presented in this work can lead to ground-state and TD electronic structure calculations where the computational scaling can be controlled, depending on the level of theory and the accuracy desired to model each one of the subsystems and their coupling.

Figure 1

A thiophene molecule interacting with a ${\mathrm{Ag}}_{20}$ pyramidal cluster. Oval and square shapes surround two possible subsystems, “A” and “B”, that the entire system can be partitioned into.

Figure 2

Sodium hydride model, where a static electric field ($E_0$) is applied along the $z$ direction.

Figure 3

Comparison between NaH binding energy curves determined by various techniques: LCOS-HF (as described in Ref. [23]), LCOS-LDA (studied in this work), and CCSD. $R$ is the internuclear distance. The curves “LCOS-HF” and “CCSD” were adapted from Ref. [23].

Figure 4

(a) Computed average of the $z$ component of the electronic position operator, in NaH, as a function of time. Upper subplot shows results for $E_0=-0.002$, and bottom subplot for $E_0=+0.002$. Purple lines: TD LCOS-LDA method; black line: TD LDA method (determined with Octopus). The TD LCOS-LDA curve is shifted to match the TD LDA line at $t=0$ [we subtract 0.48 from the original TD LCOS-LDA data, $\langle z\left(t\right)\rangle$]. (b) Same as (a), but only displaying TD LCOS-LDA curves in the range $0<t<6$. (c) Hydrogen electron population $\left[\langle N_{\mathrm{H}}\left(t\right)\rangle \right]$ as a function of time, as computed by our TD LCOS-LDA simplified model. Atomic units are implied.

Figure 5

Summary of the steps required to compute electronic properties with DFT-based LCOS methods. An additional implicit step is the selection of ground-state DFT formalism: the method based on reference Hamiltonians (Sec. 4b) or energy functionals (Sec. 4c).