Two-component linear-response time-dependent density functional theory (TDDFT) provides a unified framework that encompasses noncollinear excitations in noncollinear reference states, as well as both spin-conserving and spin-flip excitations in collinear reference states. In this work, we present a general formalism for evaluating the expectation value $\langle \hat{S}^2 \rangle$ of electronically excited states obtained within two-component TDDFT. We then derive and analyze specialized forms of the resulting equations for collinear reference determinants, for which the two-component formalism decomposes into conventional spin-conserving and spin-flip TDDFT. The resulting working equations are systematically compared with previously proposed theoretical approaches. On the basis of our analysis, $\langle \hat{S}^2 \rangle$ in the excited states is shown to arise from two distinct sources: (i) $\langle \hat{S}^2 \rangle_0$ in the reference state and (ii) additional $Δ\langle \hat{S}^2 \rangle$ introduced by the excitation process itself. Finally, we evaluate the expectation value $\langle \hat{S}^2 \rangle$ by performing two-component TDDFT calculations based on two-component DFT, unrestricted Kohn-Sham (UKS), and restricted open-shell Kohn-Sham (ROKS) reference states, respectively.
Linear-response TDDFT is prevalent due to its balance between efficiency and accuracy. 1 In the non-relativistic limit, two-component TDDFT is an adequate framework even in a noncollinear magnetic system. 2 On a collinear reference state, two-component TDDFT is divided into spin-conserving and spin-flip TDDFT. Spin-flip TDDFT, formulated on an open-shell reference state, has been demonstrated to be particularly useful for the description of conical intersections, charge-transfer excited states, nonadiabatic derivative couplings, spin-orbit couplings, and bond dissociation processes. 3 In order to further enable applications in spectroscopy, it is vital to know the spin multiplicity of excited states; that is, to calculate ⟨ Ŝ2 ⟩. However, calculating ⟨ Ŝ2 ⟩ is not trivial and hasn't been systematically explored in two-component TDDFT.
Maurice and Head-Gordon 4 derived analytical expressions for the expectation value of the total spin-squared operator, ⟨ Ŝ2 ⟩, within the framework of spin-conserving configuration interaction singles (CIS). Ipatov et al. 5 subsequently obtained expressions for ⟨ Ŝ2 ⟩ in spinconserving TDDFT. Myneni and Casida 6 addressed and resolved several ambiguities associated with the evaluation of ⟨ Ŝ2 ⟩ in spin-conserving TDDFT and proposed an ab initio formulation. Building on the conceptual framework introduced by Ipatov et al., Li et al. 7 derived the corresponding working equations for spin-flip TDDFT, which they reported in the appendix of their work.
To enrich and systematize the theoretical framework in this area, we introduce a unified formulation for the spin-squared expectation value, ⟨ Ŝ2 ⟩, within two-component time-dependent density functional theory (TDDFT). We begin from the second-quantized representation of Ŝ2 and demonstrate that its algebraic structure closely parallels that of the second-quantized electronic Hamiltonian Ĥ. On this basis, we derive explicit expressions for ⟨ Ŝ2 ⟩ evaluated over general Slater determinants, which are found to be analogous to the corresponding energy expectation values derived from the Slater-Condon rules. Subsequently, we formulate two-component TDDFT in the second-quantized equation-of-motion (EOM) framework, which is formally equivalent to the Casida equation in the first-quantized matrix representation. By substituting Ĥ with Ŝ2 in the EOM-like TDDFT expressions, we obtain a Casida-type eigenvalue equation in the first-quantized representation for the evaluation of spin-squared expectation values. In close analogy to the electronic excitation energies ω (∆⟨ Ĥ⟩), the changes in spin-squared expectation values, ∆⟨ Ŝ2 ⟩, can be computed within this Casida-like formalism. We then specialize the treatment to collinear reference states and derive further simplified working equations, which enable direct comparison with previously proposed spin-analysis schemes. Finally, we present numerical results for a range of molecular systems to illustrate and benchmark the proposed methodology.
We use Γ, Λ, Θ… for two-component AO basis functions, P, Q, R… for unspecified twocomponent molecular orbitals, and I, J, K… / A, B, C… for occupied / virtual orbitals. We use g µ , g ν , … to denote the spatial part of AO basis functions. The corresponding spin-up basis is denoted as:
And the spin-down basis is denoted as:
Also, we use the corresponding lowercase letters to denote the spin-up or spin-down parts of two-component orbitals:
We denote the column p of C α as C α p , and that of C β as C β p . We use short notations for double-electron integrals:
⟨P Q|RS⟩ = (P R|QS)
We denote the matrix representation of a one-electron spin operator ŝu as:
Here, u = x, y, z. σu is the Pauli matrix along the u-axis.
Because ŝu is a one-body operator, its second-quantized form is:
The total spin of the electrons is defined as:
By substituting the ŝu , we obtain:
There are two basic anti-commutation relations:
Using the two relations, we obtain:
We denote:
Based on the definition, we have:
Finally, we obtain:
We denote |X⟩ as the reference single slater determinant wavefunction |Ψ⟩ 0 , constructed by a set of orthonormal molecular orbitals. We usually obtain |Ψ⟩ 0 using the two-component Hartree-Fock method. A singly excited wavefunction is denoted as |Y ⟩ = â † A âI |X⟩. A doubly excited wavefunction is denoted as |Z⟩ = â † A â † B âJ âI |X⟩. Based on the Slater-Condon rule, 8,9 we can calculate the expectation value of Ŝ2 for different determinants.
Here, N is the number of electrons. We now need to further rewrite W P QRS with twocomponent orbital indices to W pqrs with one-component orbital indices. We denote the overlap matrix as:
By expanding a molecular orbital into a set of basis functions, we have:
Then,
By introducing eq. 24 into eq. 19, we produce the same result as eq. ( 19) in Ref. 10.
But we need to derive eq. 19 under a common AO representation for more convenient implementation. We define the first-order reduced densi
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