Linearized Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a post-SCF multireference method that has been successful at computing ground-and excited-state energies. However, MC-PDFT is a single-state method in which the final MC-PDFT energies do not come from diagonalization of a model-space Hamiltonian matrix, and this can lead to inaccurate topologies of potential energy surfaces near locally avoided crossings and conical intersections. Therefore, in order to perform physically correct ab initio molecular dynamics with electronically excited states or to treat Jahn-Teller instabilities, it is necessary to develop a PDFT method that recovers the correct topology throughout the entire nuclear configuration space. Here we construct an effective Hamiltonian operator, called the linearized PDFT (L-PDFT) Hamiltonian, by expanding the MC-PDFT energy expression to first order in a Taylor series of the wave function density. Diagonalization of the L-PDFT Hamiltonian gives the correct potential energy

surface topology near conical intersections and locally avoided crossings for a variety of challenging cases including phenol, methylamine, and the spiro cation.Furthermore, L-PDFT outperforms MC-PDFT and previous multi-state PDFT methods for predicting vertical excitations from a variety of representative organic chromophores.

Introduction
Understanding and properly modeling excited-state dynamics is important for many photoinduced processes in chemistry and biochemistry including photochemistry, 1,2 light-harvesting, [3][4][5] photocatalysis, [6][7][8] photosensing, 9 vision, 10,11 DNA photostability, 12 and nonadiabatic electron transfer. 13Multiconfiguration pair-density functional theory 14 (MC-PDFT) is a post-SCF method that that has been shown to be a computationally efficient method for computing potential energy surfaces (PESs) of excited states. 15,16Starting from a multiconfigurational wave function, such as that provided by the complete active space SCF (CASSCF) method, MC-PDFT computes a corrected energy through a nonvariational energy expression which is a functional of the electron density (ρ) and on-top pair density (Π).Consequently, the MC-PDFT energies are not eigenvalues of any particular Hamiltonian or quantum operator, and the energy of each state is computed independently of the other states involved in the calculation.MC-PDFT and its hybrid counterpart (HMC-PDFT) 17 have been shown to perform similarly to the much more expensive n-electron valence perturbation theory (NEVPT2) 18 for over 300 vertical excitations in the QUESTDB database. 16en modeling photochemistry and photodynamics, one frequently encounters locally avoided crossings and conical intersections: regions in which states of the same spin symmetry interact strongly with each other.This necessitates the use of an electronic structure method that includes state interaction to prevent unphysical PES crossings near these strong coupling geometries.Such methods are called multi-state methods, 19 and they include multistate 20,21 and quasi-degenerate [22][23][24] perturbation theory as well as multireference configuration interaction. 19,25They are necessary for a proper treatment of nonadiabatic dynamics or (L-PDFT) in which we express the Hamiltonian ( ĤL−PDFT ) in second quantization as an operator that is a functional of the density and pair density.We construct ĤL−PDFT by expanding the MC-PDFT energy functional in a power series of ρ and Π variables about their state-averaged values within the model space and truncate this series at first order, such that for any state |I⟩, ⟨I| ĤL−PDFT |I⟩ is a linear approximation to its MC-PDFT energy.
By construction, ĤL−PDFT is a well-defined linear operator whose off-diagonal elements are generally nonzero, and diagonalization of ĤL−PDFT within a given subspace yields a set of PES with the correct topology near conical intersections and locally avoided crossings.
Here we show that L-PDFT yields similar PES topology to CMS-PDFT for a variety of challenging cases including phenol, methylamine, and the spiro cation and, it does not have the same intrinsic limitations of CMS-PDFT for linear systems with degenerate 1 ∆ u states (see Section 4.6).Additionally, we compute the vertical excitation energy for a small test set of representative organic chromophores and find that L-PDFT is more accurate than MC-PDFT and CMS-PDFT.

Notation
Capital letters I, J label general many-electron states.Lowercase letters p, q, r, s, t, u label general spatial molecular orbitals.Repeated indices are summed implicitly.Boldfaced characters represent tensors (vectors, matrices, etc.).

Multiconfiguration Pair-Density Functional Theory (MC-PDFT)
MC-PDFT is a post-SCF method that computes the energy of a state using a functional expression similar to Kohn-Sham density functional theory (KS-DFT).The MC-PDFT energy expression for some state |I⟩ is defined as where h nuc is the nuclear-nuclear repulsion term, h pq and g pqrs are the 1-and 2-electron integrals, D I pq and d I pqrs are elements of the spinless reduced 1-and 2-particle density matrices of state |I⟩ (D I and d I respectively), and E ot is an on-top density functional of the density, ρ I , and on-top density, Π I . 14Summing all terms except the on-top functional equals the sum of the kinetic energy and classical electrostatic energy for the state.
Given a set of spatial molecular orbitals, ϕ p (r), both ρ I and Π I can be expressed as functions of D I and d I : Thus, the MC-PDFT energy expression (Eq. 1) can be expressed as an explicit function of the 1-and 2-particle density matrices for a given state:

Linearized-PDFT (L-PDFT)
The classical electrostatic and on-top functional terms in Eq. 1 are nonlinear in the densities.
The Coulomb term is quadratic with respect to D I , and in general, the on-top functional is even more nonlinear in the densities.This means that it is generally not possible to find an operator Ô such that Here, δ pq,rs is the standard Kronecker delta.Simplification and using the permutation symmetry of the 2-electron integrals allows us to write the Coulomb term as where is the Coulomb interaction with the zeroth-order density.Expanding the on-top functional term to first order allows us to write Note that both ∇ D E ot and ∇ d E ot have been defined in previous MC-PDFT analytic nuclear gradient papers 30,31 and are called the 1-and 2-electron on-top potentials, respectively, and they are given by Hence, the first-order Taylor expansion of the MC-PDFT energy of state |I⟩ around the zeroth-order densities D 0 , d 0 is given by where we have collected all constant terms (those that do not depend on D I or d I ) as Defining ĤL−PDFT as with Êpq and êpqrs being the 1-and 2-electron excitation operators respectively, yields a quantum operator that satisfies the condition of Eq. 5.
There are several different zeroth-order densities that one can expand around.For example, one might consider expanding about the Hartree-Fock densities or the ground-state densities.However, these densities would treat the ground state on a different footing than the excited states, which is undesirable.Instead, we use the state-averaged densities given by where w I satisfy and would typically be the same weights as used in a reference SA-CASSCF calculation.
These densities are desirable as they will treat all the states on an equal footing.
Letting U be the particular subspace of the Hilbert space H such that U is the span of {|I⟩} (in this paper, we take U to be the model space which is spanned by the SA-CASSCF states, as is customary in multi-state perturbation theory and in earlier MS-PDFT methods), then for equal weights, the state-averaged densities maintain the property that they are independent of the reference state basis {|I⟩}, making D 0 and d 0 functionals of the subspace U .Hence ĤL−PDFT , along with its eigenvalues and eigenvectors, are independent of the initial basis {|I⟩} and instead are functionals only of the subspace U that {|I⟩} span.That is, for each subspace U ⊆ H there is a unique quantum operator ĤL−PDFT [U ].
Whereas in KS-DFT and MC-PDFT the energy is expressed as a functional of the density of a given state, we have defined an effective Hamiltonian that is a functional of a state-averaged density within a given subspace.
Note that although the effective Hamiltonian is a functional of U , its definition is not restricted to U : a matrix element of Eq. ( 13) could be evaluated for any arbitrary pair of many-electron states in H .However, for states outside of U , we do not expect Eq. (

Hybrid Linearized-PDFT
HMC-PDFT 17 extends the original Becke concept of KS-DFT hybrid functionals 32 by using a weighted average of the CASSCF and MC-PDFT energies: where λ controls the fraction of CASSCF energy (E CAS

I
) which is included in the hybridization.We refer to hybrid calculations by using the language of hybrid functionals, for example we refer to using Eq.16 with λ equal to 0.25 and the tPBE functional in the MC-PDFT term as an HMC-PDFT calculation with the hybrid functional tPBE0.Because it has been found that hybrid functionals, such as tPBE0, are more accurate than non-hybrid ones for certain systems, 16 we also construct a hybrid linearized PDFT (HL-PDFT) Hamiltonian.
Since E CAS I is linear with respect to the densities, Taylor expanding Eq. 16 and extracting the effective HL-PDFT Hamiltonian ( ĤHL−PDFT ) yields where Ĥel is the usual electronic Hamiltonian.

Computational Details
We study the potential energy curves and PESs of several test systems that have been studied in prior MS-PDFT papers. 27,28For all cases considered here except acetylene, CMS-PDFT has been shown to perform similarly to the more expensive XMS-CASPT2; hence, we use CMS-PDFT as a benchmark for those systems.For acetylene, we use XMS-CASPT2 as our benchmark because this system has not been studied previously with any MS-PDFT method.System-specific computational details including symmetry, basis set, number of states, active space electrons and orbitals, and reaction coordinates scanned are summarized in Table 1.For the spiro cation, XMS-PDFT data is taken from the supporting information of Ref 28.
We also compute vertical excitations for a variety of organic chromophores.All of these systems were included in a prior comparison between MC-PDFT and CASPT2. 15We only consider singlet-singlet excitations in this work.We use the jul-cc-pVTZ basis set 33,34,[40][41][42] for all valence excitations, the aug-cc-pVTZ basis set 33,34 for water, the 6-31+G** basis set [35][36][37] for p-nitroaniline (pNA) and 4-(dimethylamino)benzonitrile (DMABN), and the aug-cc-pVDZ basis set 33,34 for the donor-acceptor complex of benzene (B) and tetracyanoethylene (TCNE).All vertical excitations are calculated using the SA-CASSCF ground-state geometry and were performed using C 1 symmetry.Additional information regarding the active space for each system is listed in Table 2. (commit dd0d9f1b154).Geometry optimizations were performed with the geomeTRIC 47 plugin (version 1.0) for PySCF.The XMS-PDFT and XMS-CASPT2 calculations for acetylene were performed in OpenMolcas (Version 22.10) 48 (tag 462-g00b34a15f).All PDFT calculations used the tPBE functional.In PySCF a numerical quadrature grid size of 6 (80/120 radial and 770/974 angular for atoms of period 1/2 respectively) was used.In OpenMolcas, the 'ultrafine' numerical quadrature grid (99 radial shells and 590 angular points for each atom, and a crowding factor of 10 and a fade factor of 10 for pruning angular grids) was used.
For XMS-CASPT2, no ionization-potential-electron-affinity (IPEA) 49 shift was used, but an imaginary level shift of 0.3i was used.All L-and HL-PDFT calculations used the model space spanned by the SA-CASSCF eigenvectors to construct ĤL−PDFT .All HMC-PDFT and HL-PDFT calculations used λ = 0.25 to correspond with the tPBE0 functional.

LiF
1][52][53][54][55][56][57][58][59] Prior studies have shown that CMS-PDFT gives similar potential curves to XMS-CASPT2, 28 whereas MC-and HMC-PDFT display unphysical state crossings between 4 and 6 Å as well as a large dip in the potential curve for both states 26,27 (Figure 1).As can be seen in Figure 1, L-PDFT and HL-PDFT give the correct potential curve shape as well as the correct state ordering in this region, and neither method has the large dip in the potential curve at 4 Å.
CMS-PDFT is known to perform well for this system, getting close to XMS-CASPT2, 28 while MC-PDFT has an unphysical state crossing at 10 Å and a dip in the two highest states 27 (Figure 2).The figure shows that the L-and HL-PDFT calculations do not have dips in their potential curves at large Li-H internuclear distances, but they suffer from
CMS-PDFT has shown to give potential curves similar to those calculated by the much more expensive XMS-CASPT2 for all of these pathways. 28The eclipsed geometry dissociation pathway passes very close to the conical intersection between the two states; whereas the 90 • and 100 • dissociations are further from the conical intersection yielding larger gaps
We consider two phenol photodissociations paths that differ in the C-C-O-H dihedral angle: φ = 1 • and 10 • .Figure 5 shows that L-and HL-PDFT are in good agreement with CMS-, MC-, and HMC-PDFT.The L-and HL-PDFT curves, like CMS-PDFT, have physically reasonable locally avoided crossings near the conical intersection along both photodissociation pathways.

Spiro Cation
The 2,2 ′ ,6,6 ′ -tetrahydro-4H,4 ′ H-5,5 ′ -spirobi[cyclopenta-[c]pyrrole] cation, which we call the spiro cation, is a mixed-valence molecule with two subsystems that share a central, bridging carbon.Due to the Jahn-Teller effect, the cation hole is either partly localized on the left or right ring.The spiro cation structure is shown in Figure 6.Letting R (1) and R (2) denote the coordinates where the hole is localized on the left and right respectively, we consider a linear synchronous path from structure (1) → structure (2), and structures along the path given by where ξ is a unitless reaction coordinate.The spiro cation is at an equilibrium geometry when ξ = ±0.5.For ξ = 0, an average of the two equilibrium structures is obtained, and this can be understood as the midpoint structure for intramolecular charge transfer.It has previously been shown that multireference perturbation theory can reasonable describe the PES only when going to the third order, 39,99,100 and this shows that the spiro cation a very difficult system to treat.CMS-PDFT has been shown to predict the correct PES topology, 28 whereas XMS-PDFT and MC-PDFT fail in that they produce unphysical dips at ξ = 0 27 (Figure 7a).The L-PDFT and HL-PDFT potential curves are shown in Figure 7. Like CMS-PDFT, these methods do not have a dip at the high symmetry point ξ = 0, as is present in the MC-PDFT, HMC-PDFT, and XMS-PDFT potential curves 27 (Figure 7).In fact, L-PDFT and HL-PDFT hardly differ at all from CMS-PDFT, even at the high-symmetry point, giving good quantitative agreement between the methods.Furthermore, the coupling term between these two states (H L−PDFT

01
) goes to zero as the molecule approaches the midpoint structure (ξ → 0) (Figure S2), indicating that ĤL−PDFT maintains the correct physical charge transfer symmetry at that point.

Acetylene (C 2 H 2 )
We now consider acetylene, a linear system with degenerate 1 ∆ u excited states at its groundstate equilibrium geometry.We consider the four lowest valence excited singlet states of acetylene as a function of one of the C-C-H bond angles (defined to be θ where θ = 180 • is the linear geometry (the other C-C-H angle is kept linear).An introduction to the energetics, symmetries, and geometry dependence of these states is provided in articles by Cui et al. 101 and Ventura et al. 102 The symmetries of these states for planar, nonlinear geometries are 1 A ′ , 1 A ′′ , 1 A ′ , and 1 A ′′ , in order of increasing energy.For linear geometries, they become and two components of a 1 ∆ u state.Our goal here is to test the accuracy of L-PDFT on this system for which CMS-PDFT is known to provide an incorrect description of the degenerate 1 ∆ u states at θ = 180 • .
Our CMS-PDFT calculations are unable to compute the energy at θ = 180 • due to the intermediate-state optimization not converging, so we omit this point and only consider θ between 135 • and 179 • .Figure 8 shows the three excited singlet states for all the methods considered, with the zero of energy set to the first excited state at θ = 179 • .The upper states should become doubly degenerate 1 ∆ u states as θ → 180 • , and that is clearly seen for XMS-CASPT2 and XMS-PDFT, although the upper 1 ∆ u potential of XMS-PDFT has the wrong curvature around 160 • .The CMS-PDFT 1 ∆ u states, however, incorrectly remain split as θ → 180 • (Figure 8a).The ground-state potential curves are almost identical for all the methods except for CMS-PDFT, which has a discontinuity at θ = 179 • (Figure S3).
This discontinuity likely results from the structure being too close to the linear structure, and thus the intermediate state optimization is struggling.
Figure 8: Potential energy curves of the three lowest excited singlet states of acetylene (the states that become 1 Σ − u and two components of 1 ∆ u at linear geometries) computed with XMS-CASPT2, XMS-PDFT, CMS-PDFT, MC-PDFT, HMC-PDFT, L-PDFT, and HL-PDFT.The zero of energy is taken as the energy of the first excited singlet state at θ = 179 • .The ground-state potential energy curves for these methods are plotted in Figure S3.
Figure 8b shows that L-PDFT and MC-PDFT PESs are in excellent agreement with XMS-CASPT2; correctly reproducing the degenerate 1 ∆ u states as ξ → 0. Figure S4 shows the excited states of all the methods with the zero of energy set to the ground state minimum energy to highlight the difference in vertical excitation energies from the ground state.That figure shows that all the methods produce different vertical excitation energies such that all the PDFT methods except CMS-PDFT have lower excitation energies for 1 Σ − u and 1 ∆ u states than does XMS-CASPT2.The CMS-PDFT method, despite its inability to recover the degenerate 1 ∆ u states, yields vertical gaps in good agreement with XMS-CASPT2 except for the uppermost 1 ∆ u state.

Vertical Excitation Benchmarking
Here we test the quantitative accuracy of L-PDFT vertical excitation energies for a variety of organic chromophores which have been studied previously with MC-PDFT and CASPT2. 15r test set includes the lowest-energy spin-conserving valence excitation energies of 13 organic molecules (acetaldehyde, acetone, formaldehyde, pyrazine, pyridine, pyrimidine, stetrazine, ethylene, butadiene, benzene, naphthalene, furan, and hexatriene).We also test one Rydberg state: the lowest singlet excitation of water.In addition, we also include energies for pNA and DMABN intramolecular charge transfer excitations, as well as the B-TCNE intermolecular charge transfer excitation.L-PDFT performs the best of all the methods for valence excitations with a MUE of 0.24 eV.
For the Rydberg excitation of water, all of the methods perform similarly, with almost no error in the vertical excitation energy in comparison to the reference value.
Finally, for the charge transfer excitations, CMS-PDFT again performs worse than MC-PDFT for these test systems with a substantially higher MUE of 0.27 eV whereas L-PDFT and MC-PDFT have very small MUEs of 0.09 and 0.08 eV respectively.HL-PDFT has a slightly worse MUE of 0.11 eV.The performance of HMC-PDFT is between that of HL-PDFT and CMS-PDFT.
If we consider all of the excitations, L-PDFT performs the best with a MUE of 0.20 eV followed by MC-PDFT with a MUE of 0.24 eV.CMS-PDFT performs worse than both MC-PDFT and L-PDFT for these vertical excitations with an overall MUE of 0.39 eV.For both pyrimidine and s-tetrazine, CMS-PDFT agrees with CASSCF, which performs poorly at describing these transitions (Table 3).We see that L-PDFT corrects the MC-PDFT vertical excitation for ethylene (MC-PDFT greatly underestimates this excitation).In most cases though, L-PDFT yields similar excitation energies to MC-PDFT, although it is still unknown how L-PDFT will perform when considering a larger number of states (N > 5), and when the number of states whose density matrices are averaged in the of the model space grows.Regardless, it is very encouraging that on this small but representative set of organic molecules, L-PDFT performs as well as MC-PDFT at predicting the vertical excitations.

Conclusion
In contrast to previous MS-PDFT methods that define the diagonal and off-diagonal effective ĤL−PDFT [U ] is constructed such that its expectation value for any state yields a linear approximation to the MC-PDFT energy of that state around the state-averaged densities.
Diagonalization of ĤL−PDFT [U ] yields PESs that have the correct topology near conical intersections and locally avoided crossings, a property that is also enforced in other MS-PDFT methods such as CMS-PDFT.Unlike XMS-PDFT, L-PDFT is able to correctly reproduce the PES for the difficult spiro cation.Additionally, L-PDFT does not fail for degenerate states of nearly-linear molecules as CMS-PDFT does.Benchmarking L-PDFT vertical excitation energies for a representative set of organic chromophores showed that L-PDFT performs similarly to MC-PDFT, whereas CMS-PDFT performs worse than MC-PDFT.Further, we note that the computational cost of L-PDFT scales as a constant with the number of states included in the state averaging, whereas MC-PDFT and CMS-PDFT scale linearly with the number of states, making L-PDFT formally a faster method.Lastly, L-PDFT also has the ) to be adequately satisfied, since the densities of those states did not contribute to the average density which defines the operator.Therefore, in this work we project ĤL−PDFT [U ] into the same subspace U and diagonalize to yield a particular set of eigenvectors ({|M ⟩}) and eigenvalues ( E L−PDFTM).Since the {|M ⟩} come from diagonalization of a Hermitian operator, they should provide the correct PES topology near conical intersections and locally avoided crossings.As an anticipation of future work, we note that L-PDFT should provide simpler expressions than CMS-PDFT for analytic gradients and other response properties (such as excited-state dipole moments, transition dipole moments, and nonadiabatic coupling vectors) because the final L-PDFT states are not dependent on a nonvariationally-optimized intermediate state basis (as in CMS-PDFT).

Figure 2 :
Figure 2: Potential energy curve for the 3 upper 1 Σ states of LiH computed with CMS-PDFT, MC-PDFT, HMC-PDFT, L-PDFT, and HL-PDFT.The ground state for each method is plotted in Figure S1.

Figure 3 :
Figure 3: The lowest energy structure of methylamine for each torsional angle studied.

Figure 4
Figure4shows that that both L-and HL-PDFT are in excellent agreement with CMS-PDFT in the region of strong coupling for all three dihedral angles.Near the equilibrium structures (near the minimum of the ground-state PES), HL-PDFT is in good agreement with CMS-PDFT whereas L-PDFT differs.

Figure 5 :
Figure 5: Potential energy curve of the two lowest states of phenol computed with CMS-PDFT, MC-PDFT, HMC-PDFT, L-PDFT, and HL-PDFT with a fixed dihedral φ.There is a gap in the potential energy curve due to an avoided crossing with a third state.(a,b) φ = 1 • .(c,d) φ = 10 • .

Table 1 :
Systems for which potential curves are studied along with the symmetry, basis set, number of states (N ), number of active space electrons (n e ), active space MOs used, and the internal coordinates scanned.

Table 3
summarizes the vertical excitation energies for all of the systems.CASSCF, which lacks external correlation (dynamic correlation outside of the active space), performs poorly compared to the other methods, with an overall mean unsigned error (MUE) of 0.57 eV.Considering the valence excitations first, CMS-PDFT has a larger MUE as compared to MC-PDFT, and the hybrid methods HMC-and HL-PDFT perform similarly to MC-PDFT.