We derive analytical nuclear gradients for state-averaged configuration interaction singles (SACIS) and its spin-projected extension (SAECIS), enabling efficient geometry optimization and minimum-energy conical intersection (MECX) searches within a low-cost CIS-based framework. The formulation employs a Lagrangian approach and explicitly removes null-space contributions in the coupled perturbed equations to ensure numerically stable gradients. For twisted-pyramidalized ethylene, both SACIS and SAECIS qualitatively reproduce the correct conical intersection topology, in sharp contrast to conventional CIS and ECIS. Benchmark calculations on twelve MECXs demonstrate that both methods reproduce geometries with mean RMSDs below 0.1~Å relative to high-level reference methods. SACIS captures the essential degeneracy through variational orbital relaxation, which alleviates ground-state Hartree--Fock (HF) orbital bias and effectively incorporates static correlation through localization effects; notably, spin projection is found to be non-essential for the qualitative description of these intersections. Overall, SACIS and SAECIS provide qualitatively reliable CX descriptions at mean-field computational cost in a black-box manner. Given their comparable accuracy and the additional overhead associated with spin projection, SACIS offers a more favorable cost-performance balance for general applications, whereas SAECIS may become advantageous when higher excited states with significant double-excitation character are involved.
Conical intersections (CXs) are fundamental features of molecular potential energy surfaces, providing efficient pathways for radiationless transitions between electronic states. 1,2 They govern ultrafast internal conversion, photoisomerization, and nonadiabatic reaction dynamics in a wide range of photochemical and photophysical processes. [3][4][5][6] Because CXs arise from degeneracies between electronic states as functions of nuclear coordinates, their reliable theoretical description requires a balanced treatment of multiple states on an equal footing. Multireference wave function methods, most notably state-averaged complete active space self-consistent field (SA-CASSCF), provide a rigorous framework for describing CXs, through variational orbital optimization and explicit state averaging. [7][8][9] However, their steep computational cost and sensitivity to active-space selection significantly limit their applicability to larger systems. These limitations have motivated the development of computationally less demanding alternatives. [10][11][12][13] Configuration interaction singles (CIS) offers an appealing mean-field-level approach to excited states due to its conceptual simplicity and low computational cost. Nevertheless, CIS is intrinsically incapable of describing CXs because it lacks the static correlation required to treat near-degenerate electronic states consistently. [14][15][16] This deficiency is shared by linear-response timedependent density functional theory (TDDFT) 17,18 and its Tamm-Dancoff approximation (TDA), 19 whose failure near CXs has been extensively documented. 16,20-22 a) Electronic mail: tsuchimochi@gmail.com Spin-flip (SF) approaches, including SFCIS 23,24 and SF-TDDFT, 25 partially remedy this problem by accessing multiple low-spin states from a high-spin reference. Because spin-flip excitations span configurations corresponding to different electronic characters, these methods can reproduce the characteristic topology of CXs in many cases. 26,27 An alternative strategy for recovering static correlation is spin projection. [28][29][30] By projecting broken-symmetry determinants onto proper spin eigenstates, spin projection restores spin symmetry and improves the qualitative description of strongly correlated systems. [31][32][33] When combined with correlated treatments, it can approach quantitative accuracy. [34][35][36][37] Within CIS-based frameworks, this concept led to time-dependent projected Hartree-Fock and spin-extended CIS (ECIS) to treat excited states. 38 Although ECIS improves descriptions of excited states in strongly correlated regimes, it remains a linear-response method based on a single reference state and is therefore not designed for a balanced multistate treatment required at CXs. 39 The underlying difficulty in CIS, TDDFT, and related approaches can also be traced to strong ground-state orbital bias. Orbitals optimized for the closed-shell ground state are generally ill-suited for excited states with qualitatively different electronic character such as chargetransfer states 40,41 , core excited states, 42 and Rydberg states. 43 Near a CX, where two states differ dramatically in electronic structure and become nearly degenerate, balanced orbital relaxation also becomes essential.
This perspective motivates state-averaged orbital optimization within a CIS framework. Recently, stateaveraged formulations of orbital-optimized CIS (SACIS) and its spin-projected extension, state-averaged ECIS (SAECIS), were introduced. 43 By optimizing orbitals with respect to a state-averaged objective, these methods remove preferential bias toward any single state and provide a variational description of multiple interacting states at mean-field cost. Deliberate spinsymmetry breaking followed by projection in SAECIS introduces additional variational flexibility, effectively allowing access to singlet configurations with partial double-excitation character as previously observed in ECIS. 38 While SACIS and SAECIS show promising qualitative behavior, practical applications to geometry optimization and minimum-energy CX (MECX) searches require analytical nuclear gradients. Formally, their derivation is straightforward and follows the standard Lagrangian framework. [44][45][46] Analytical derivatives for spinprojected methods have also been formulated. 47,48 In the state-averaged framework of SACIS and SAECIS, the first derivative of the Lagrangian involves the electronic Hessian, which can be constructed using our previous derivations. 41,43 However, redundant parametrization in these methods renders the electronic Hessian singular, leading to nonunique solutions in the associated coupled perturbed equations. Without careful treatment, this null space contaminates the Z-vector solution and the resulting gradients. In this work, we derive analytical nuclear gradients for SACIS and SAECIS and introduce an explicit projection procedure to eliminate null-space contributions, ens
This content is AI-processed based on open access ArXiv data.