Introduction

Kohn-Sham density functional theory (KS-DFT) 1 has been successful in treating many chemical problems, but it is less accurate for treating inherently multi-congurational electronic stateswhich are called strongly correlated statesthan for treating states well represented by a single Slater determinantwhich are called weakly correlated. 2 Strong correlation usually arises from near degeneracy of two or more states, and excited electronic states are usually strongly correlated, and oen they strongly interact with other states. Thus, the accurate treatment of strongly correlated states is necessary for spectroscopy and photochemistry. 3,4 Furthermore, the accurate treatment of strongly correlated sets of states is also required to properly describe magnetic effects. 5,6 Although KS-DFT has lower accuracy for strongly correlated states than for weakly correlated ones, for large molecules it is much less expensive than wave function theory (WFT) methods of comparable accuracy. We have proposed multi-conguration pair-density functional theory (MC-PDFT) as a method that builds on a multi-congurational self-consistent-eld (MCSCF) reference wave function and is more innately suitable for strongly correlated systems than KS-DFT; MC-PDFT also has the advantage of being computationally less expensive compared with WFT methods in terms of computer time and memory with comparably accurate treatments of correlation energy. 7,8 We refer the reader to a recent review article 9 that compares PDFT to other ways to combine wave function methods and density functional methods for excitedstate calculations.

When states are nearly degenerate and have the same symmetry, they interact strongly with each other, and they should be treated by a method that gives the correct topography 4 of adiabatic potential energy surfaces (PESs) at conical intersections; such methods are called multi-state (MS) methods. For example, in WFT, multireference Møller-Plesset perturbation theory 10 is a state-specic method because it calculates the nal approximation of the energy of each state separately, whereas multi-conguration quasi-degenerate perturbation theory (MC-QDPT) 11 and extended MC-QDPT (XMC-QDPT) 12 are multi-state methods because the nal energies are eigenvalues of the same matrix (hence they interact through the off-diagonal elements of that matrix). Similarly, complete active space perturbation theory (CASPT2) 13 is a state-specic method, and multi-state CASPT2 (MS-CASPT2) 14 and extended MS-CASPT2 (XMS-CASPT2) 15 are multi-state methods.

The original MC-PDFT is a state-specic method. We recently proposed state-interaction PDFT (SI-PDFT) as a multi-state generalization; 16 SI-PDFT yields the correct topography of adiabatic PESs for conical intersections and it has been applied successfully to several problems; [16][17][18] but it is inconvenient because two MCSCF calculations and two sets of orbitals are required, and it puts the ground state on an unequal footing with the excited states, which is sometimes undesirable (for example, for treating magnetic states). In the present paper we present two new multi-state methods that eliminate these drawbacks of SI-PDFT. One is called extended-multi-state-PDFT (XMS-PDFT) because it uses the intermediate basis proposed by Granovsky 12 for XMC-QDPT, and the other is called variational-multi-state-PDFT (VMS-PDFT) because it obtains an intermediate basis by variationally maximizing the sum of MC-PDFT energies for the intermediate states. We approximate the VMS-PDFT method by using a Fourier series expansion; this method for VMS-PDFT is called the Fourier-multi-state-PDFT (FMS-PDFT) method.

A key aspect of all the above-mentioned MS methods is that they determine a model space spanned by the states to be treated as strongly interacting. Similar to XMC-QDPT or XMS-CASPT2, XMS-PDFT and VMS-PDFT build up a model space that spans the N lowest-energy states optimized in a state-averaged CASSCF (SA-CASSCF) calculation. (Generalizations to incomplete active spaces and smaller model spaces are straightforward but are not considered here.) The model space states are called the intermediate basis and are obtained by unitary transformation from the SA-CASSCF states.

Section 2 explains the two new methods and the Fourier-based approximation of VMS-PDFT. Section 3 species the computational details for several test systems, including those that were previously studied by SI-PDFT. Section 4 presents applications of the new methods to these test systems and evaluates their performances. Section 5 provides the concluding remarks.

Theory

MC-PDFT

The MC-PDFT method may be based on single-state CASSCF (SS-CASSCF) calculations or on SA-CASSCF calculations. In the present article we consider the latter type of calculation, in which case one starts with a reference wave function obtained by performing an SA-CASSCF calculation and given by

where i is the index of a conguration state function (CSF), and I is the index of a reference state. The MC-PDFT energy for state I is

where the terms are the electronic kinetic energy, the classical electrostatic energy (which is the sum of the nuclear-nuclear repulsion, the electron-nuclear attraction energy, and the classical electron-electron repulsion), and the on-top energy computed as a functional of the density r I and the on-top density P I , both computed from |J I i, with the latter given by P I ðrÞ ¼ ð J * I ðr 1 ; r 2 ; .; r Ne ÞJ I ðr 1 ; r 2 ; .; r Ne Þdr 3 .dr Ne | r 1 ¼r 2 ¼r :

Eqn (2) applies to MC-PDFT calculations starting with either SS-CASSCF or SA-CASSCF. We note that it does not separate the energy into an uncorrelated component, a static correlation component, and a dynamic correlation component. Because the original MC-PDFT method computes the state energies independently, it is a state-specic method in the sense that the nal energy of each state is computed separately, even if one starts with SA-CASSCF kinetic energies, densities, and on-top densities.

Multi-state MC-PDFT

To obtain the correct topography of PESs at conical intersections, we have proposed the SI-PDFT method 16 as an MS extension of MC-PDFT. In SI-PDFT, we generate a set of intermediate states with the reference SA-CASSCF states and an auxiliary state from a state-specic ground-state CASSCF calculation. The ground intermediate state is obtained by projecting the SS-CASSCF state into the space spanned by the SA-CASSCF states, and the other intermediate states are obtained by performing Schmidt orthogonalization of the excited states obtained by the SA-CASSCF calculation to the ground intermediate state. Then one constructs an effective Hamiltonian in the intermediate state basis and diagonalizes it to get the SI-PDFT energy for each state. This treats the ground and excited states unequally. Moreover, using different orbital sets (i.e., using both the orbitals from the SS-CASSCF calculation and those from the SA-CASSCF calculation) is inconvenient.

To avoid these problems, we next propose two new multi-state MC-PDFT methods that use only one set of orbitals.

In general, the intermediate states are obtained by a unitary transformation:

where |F I i is an intermediate state, and |J J i is an SA-CASSCF state. The electronic Hamiltonian of the molecule is diagonal in the SA-CASSCF states but usually not in the intermediate basis.

We construct an effective Hamiltonian in the intermediate-state basis with diagonal elements dened as

where E MC-PDFT I is the MC-PDFT energy for the intermediate state |F I i. The offdiagonal elements of the effective Hamiltonian are dened as

with I, J ¼ 1, 2, ., N, where N is the number of states in the model space. (In the present work, the number of states in the model space is always the same as the number of states averaged in the SA-CASSCF calculation.) The effective Hamiltonian is then diagonalized to give the multi-state MC-PDFT energies for each adiabatic state. Following the above scheme, we next introduce two strategies (XMS-PDFT and VMS-PDFT) to generate the matrix U, yielding U X and U V , respectively.

XMS-PDFT

The intermediate basis in XMS-PDFT diagonalizes the effective Hamiltonian suggested by Granovsky for XMC-QDPT in ref. 12, where he stressed that "the effective Hamiltonian should be a function of the subspace spanned by the selected CI vectors, rather than a function of any particular choice of basis in this subspace" and that "the computed energies must be uniquely dened, continuous and smooth functions of the molecular geometry and any other external parameters, with possible exceptions at the manifolds of their accidental degeneracy such as conical intersections". The XMS-CASPT2 method also uses this intermediate basis. We use the XMS-CASPT2 procedure 15 to explain this, and the explanation starts by recalling the procedure for MS-CASPT2.

In MS-CASPT2, the unperturbed Hamiltonian is dened as

where

is the projection operator onto the SA-CASSCF state space and

is the projection operator onto the complementary state space. In MS-CASPT2, the state-space Fock operator is dened as

where E pq ¼ a † p a q is a single-excitation operator, a † p and a q are creation and annihilation operators on molecular orbitals p and q, respectively, and f pq is an element in the orbital Fock matrix

where h pq contains the electronic kinetic energy and electron-Coulomb interaction, d rs is a state-averaged density matrix element, and J rs pq and K rs pq are twoelectron integrals. The matrix elements of the state-space Fock matrix are dened as

The state-space Fock matrix dened in eqn (10) is not necessarily diagonal, because the reference wave functions (i.e., the SA-CASSCF wave functions) are the eigenstates of the Hamiltonian operator, not necessarily the eigenstates of the state-space Fock operator or the zeroth-order Hamiltonian.

The MS-CAPST2 method neglects the off-diagonal elements of the state-space Fock matrix, but following the prescription used in the XMS-CASPT2 method, the XMS-PDFT method diagonalizes the state-space Fock matrix by a transformation matrix U X :

The U X matrix determined this way then yields the intermediate states dened by

where F I is an intermediate state in XMS-PDFT (and also in XMS-CASPT2). With the same transformation, we get a Hamiltonian matrix in the intermediate basis,

where ) and eigenvectors are obtained by diagonalizing the effective Hamiltonian matrix.

We notice that the off-diagonal elements in the state-space Fock matrix are zero for states with different symmetries. This suggests that the XMS-PDFT method is identical to MC-PDFT if all states in the model space belong to different irreps (similarly, XMS-CASPT2 is identical to MS-CASPT2 or even singlestate CASPT2 for such a case). This is not a problem, but we have found that the off-diagonal elements in the state-space Fock matrix are almost zero for many geometries in some mixed-valence 19 systems (see Fig. S3, † where sections and gures with the prex S are in the ESI †) even when the states have the same symmetry, and we will see that XMS-PDFT does not always give good results for such systems. Next we present the VMS-PDFT method which does not have this problem (but it is more expensive).

VMS-PDFT

The trace of the effective Hamiltonian dened above is given by

Although the kinetic energy component in an MC-PDFT energy (i.e. the rst term of eqn ( 2)) is unitarily invariant, the classical electrostatic energy and the ontop energy are not, and therefore the trace in eqn ( 14) depends on the transformation matrix U. In VMS-PDFT, the transformation matrix U V that yields the intermediate basis is chosen so that this trace is maximized. Just as in XMS-PDFT, but using the new intermediate basis, VMS-PDFT then evaluates the energies by diagonalizing an effective Hamiltonian dened such that the diagonal elements are MC-PDFT energies in the intermediate basis and the off-diagonal elements are computed by standard wave function theory in the intermediate basis.

The motivation for using a transformation that maximizes the sum of on-top energies for intermediate states is a physical one, namely that the diagonalization of the effective Hamiltonian can be interpreted as adding extra correlation to the energies of intermediate states, so this correlation energy should not already be present in the diagonal elements.

At present, we do not have an analytic procedure to nd the transformation matrix U V that completely maximizes eqn (14). Instead, we propose here a numerical way to approximate the maximization in a practical and smooth way by tting eqn ( 14) to a Fourier series. We call this implementation the FMS-PDFT method.

We rst present FMS-PDFT for a two-state calculation. The unitary transformation between two states (J I ,J J ) can be parameterized as

where q IJ is the rotation angle between the two states. Applying U V IJ (q IJ ) to a pair of states yields

Now consider applying U q þ p 2 to the two states; this yields

Comparing eqn ( 16) and (17) shows that U V IJ (q IJ ) and

generate the same two states but with different ordering, and thus they give the same trace of the effective Hamiltonian matrix. This means that the trace of the effective Hamiltonian matrix has a period of p 2 , and therefore the Fourier expansion of the effective Hamiltonian can be written as

We keep only the terms with n ¼ 1 in the sum on the right hand side of eqn (18); then the equation can be parameterized as

and the unknown parameters A, B, and C can be obtained by a three-point tting.

In this paper, three values (0 , 30 , and 60 ) for q IJ are applied to determine these three parameters for each single-point energy calculation. (We use these same three angles in all cases.) The rotation angle is taken as the one that maximizes eqn (19).

For FMS-PDFT calculations with N states (where N is greater than 2), we write U V as a product of transformation matrices

where U I(I+1) (q I,I+1 ) is a unitary matrix that rotates states I and (I + 1) as in eqn (15). When each U I(I+1) is applied, we have

where I ranges from 1 to N À 1, F (I) denotes the N intermediate states aer transformation of U 12 U 23 /U I(I+1) , and F (0) denotes the initial states, which are SA-CASSCF states in this paper. Thus, eqn (19) can also be applied to tting the trace of the effective Hamiltonian for each unitary transformation U I(I+1) .

Notice that neither do we include all the N(N À 1)/2 transformation matrices U IJ (I < J) nor do we transform the states iteratively to reach the absolute maximum trace. Since strong couplings mostly occur between adjacent states, it is a reasonable and economical practice to only consider the (N À 1) unitary transformations U I(I+1) as presented above. Furthermore, we will see below that the results are already good with a single pass as in eqn (20). Furthermore, stopping with a single pass gives smoother results than one would obtain if one used a convergence criterion that led to different numbers of iterations at different geometries.

The FMS-PDFT method is also implemented for wave functions optimized with the density matrix renormalization group [20][21][22][23][24][25] (DMRG) approach; the combination of state-specic PDFT and DMRG was introduced previously 26,27 and is here extended to an MS treatment. The FMS-PDFT/DMRG method is an extension of the FMS-PDFT method described above except that it is based on an SA-DMRG 28 calculation instead of an SA-CASSCF starting point, and this requires a change in implementation since the DMRG wave function is not explicitly expanded in a CSF basis. Therefore we do not obtain intermediate states with eqn (4); instead, the MC-PDFT energies for intermediate states are calculated with the transformed one-body and two-body density matrices,

where D JK and d JK are one-body and two-body transition density matrices between the reference states J and K, and DII and dII are one-body and two-body density matrices for the intermediate state I.

Computational details

The calculations are performed in OpenMolcas v18.09, tag 548-g19e2926-dirty, 29 with codes modied to perform XMS-PDFT and FMS-PDFT calculations. The DMRG calculations are performed with the QCMaquis soware suite 25,[30][31][32] in In XMS-CAPST2 calculations, an ionization-potential-electron-affinity (IPEA) shi 33 of 0.25 a.u. is used. In the FMS-PDFT/DMRG calculations for phenol, we used the same active space as we used for regular FMS-PDFT calculations. The bond dimension (M) is set to 500. In the PDFT calculations, we used the translated PBE (tPBE) on-top functional.

Table 1 presents the wave function symmetry, basis set, number of averaged states, number of active electrons, and identities of active MOs for each system studied. The internal coordinates that are scanned for each system are shown in Table 2. The geometries are available in Section S1. †

For the HNCO calculations with bond length r(NC) ¼ 2.0-2.5 Å and torsion angle s(HNCO) ¼ 150 (discussed in Section 4.3), we carried out the VMS-PDFT calculations using a numerical maximization procedure instead of the Fourier series algorithm because the 3-point tting in FMS-PDFT fails for that limited region due to the trace of the effective Hamiltonian changing slowly with respect to the rotation angle, so that keeping only the terms with n ¼ 1 in the sum on the right hand side of eqn (18) is inadequate. In all other cases the Fourier series method proved adequate.

Results and discussion

Lithium uoride (LiF)

Lithium uoride has an avoided crossing of the ground state and rst excited state that has been widely studied. [39][40][41][42][43][44][45][46] The ground state at the equilibrium distance is ionic, corresponding to the (2p z,F ) 2 (2s Li ) 0 conguration. The ground state has A 1 symmetry in the C 2v group. This state interacts with another A 1 state that corresponds to two neutral ground-state atoms, namely (2p z,F ) 1 (2s Li ) 1 . The accurate value of the distance of the avoided crossing is about 7.4 Å. 39 However, theoretical calculations usually underestimate the distance by 1.0 Å, with an exception being the calculation in ref. 41.

The MC-PDFT method gives an unphysical double crossing between 4 Å and 6 Å, associated with a "dip" of the energy curve, as shown in Fig. 1(a) and (c). The XMS-PDFT and FMS-PDFT methods, however, remove the incorrect double crossing and also recover the expected shape of the avoided crossing at a larger distance. Additionally, the two new multi-state PDFT methods preserve the correct asymptotic character of the two states, and they work well for the whole potential energy curve. To check the tting accuracy of FMS-PDFT, we compare the trace of the effective Hamiltonian matrix obtained by 3-point tting to that obtained by non-tted calculations. In Table 3, we list the rotation angles for various Li-F bond lengths and the corresponding trace of the effective Hamiltonian obtained by calculations with and without tting. The mean unsigned error (MUE) of the trace in the tted calculation is less than 3 meV, which is much less than the intrinsic error in the method and is adequate for most applications.

Lithium hydride (LiH)

The ground state of lithium hydride is an ionic state near the equilibrium geometry, but this state interacts with three covalent states, corresponding to (2s Li ) 1 (1s H ) 1 , (2sp z,Li ) 1 (1s H ) 1 , and (3s Li ) 1 (1s H ) 1 congurations, as the Li-H bond dissociates. All four states have A 1 symmetry in the C 2v point group.

Despite the complexity of the ionic state of LiH crossing with at least three other states as shown in Fig. 2, a similar pattern to LiF is still found for the third and fourth states of LiH beyond 10 Å. The zoomed-in regions in Fig. 2

Faraday Discussions Paper

show that the MC-PDFT curves for the third and fourth states still have a dip and a double crossing, while XMS-PDFT and FMS-PDFT recover the avoided crossing of the two states and also remove the dip. The rst (red) and second (blue) states calculated by XMS-PDFT and FMS-PDFT agree very well with those calculated by XMS-CASPT2. The minimum energy separation between the third and fourth states is 0.10 eV at 10.66 Å by XMS-PDFT, 0.08 eV at 11.63 Å by FMS-PDFT, and 0.07 eV at 11.28 Å by XMS-CASPT2. The shapes of the XMS-CASPT2 curves match much better with XMS-PDFT and FMS-PDFT than with MC-PDFT, especially for the energy minima of the excited states.

The potential energy curves of XMS-CASPT2 and FMS-PDFT overlap very well, demonstrating the superiority of physically motivated FMS-PDFT, and also showing that applying only (N À1) rotations to an N-state calculation without iteration is sufficient for FMS-PDFT. Note that there is a bump around 3 Å for the FMS-PDFT potential energy curve. A similar but more indistinct bump can also be found on the XMS-PDFT potential energy curve. The bumps are a result of the interaction of the 4 th state and the next higher one, which is not included in the calculation. This is a problem not just with the methods presented here but with the potential curves calculated by any method based on SA-CASSCF; the highest included state usually has an avoided crossing with the rst unincluded state, and this causes some nonsmoothness in the potential curves.

Isocyanic acid (HNCO)

We next turn to avoided-crossing regions in polyatomics, and we remind the reader that avoided crossings along a polyatomic path are a signal that one is close to a conical intersection. 48 For planar HNCO, the rst two singlet states have A 0 and A 00 symmetry in the C s point group. Along the dissociation path of the NC bond, these two states cross each other. However, if the molecule becomes nonplanar, the crossing becomes avoided (since their coupling becomes symmetry-allowed), and the states switch character as they pass the locally avoided crossing. Based on previous work in our group, 49 we xed two bond lengths, r(HN) ¼ 1.0584 Å (2.0a 0 ) and r(CO) ¼ 1.1906 Å

(2.5a 0 ), and two bond angles, a(HNC) ¼ 110 , and b(NCO) ¼ 100 and then varied the r(NC) bond length from 1.25 to 3.00 Å and the s(HNCO) torsion angle from 180 to 130 . We present the curves when the torsion angle is 150 and 175 in Fig. 3 and the curves at other torsion angles are shown in Fig. S1. † Fig. 3 shows that the shapes of the potential curves predicted by XMS-PDFT are similar to those obtained with XMS-CASPT2, although XMS-PDFT predicts a slightly wider energy separation (about 0.17-0.22 eV) of the two states in the region of the equilibrium well and around the region of the energy barrier close to the planar geometry. Fig. S1 † shows that further from the planar geometry, with the s(HNCO) torsion angle less than 140 , the wider energy separation of XMS-PDFT still holds for the region of the energy barrier, but in the region of the equilibrium well the energy separation of XMS-PDFT becomes slightly narrower than that of XMS-CASPT2. Despite these minor differences between the curves of the two methods, both methods are successful in showing the avoided crossing. Close to the equilibrium geometry, XMS-PDFT places the avoided crossing at an N-C distance about 0.1 Å shorter than that of XMS-CASPT2. Since the two states obtained with XMS-PDFT have a slightly wider separation, the avoidance of the two states is more obvious than the avoidance with XMS-CASPT2. Further from the planar geometry, the crossing point of the two states moves to a shorter N-C distance, and we nd that the two states still avoid each other smoothly.

A torsion angle of 150 is chosen to test the performance of the FMS-PDFT method. A similar avoided crossing around 1.7 Å is observed for both FMS-PDFT and XMS-CASPT2. Although the separation of the two states in the region of 2.0-2.5 Å is still wider for VMS-PDFT compared with XMS-CASPT2, VMS-PDFT is shown to be well-behaved, even though we used a combination of FMS-PDFT and numerical VMS-PDFT in this case, as discussed in Section 3.

Methylamine (CH 3 NH 2 )

The potential energy surfaces, dynamics, and spectroscopy of CH 3 NH 2 have been widely studied both experimentally and theoretically. 18,[50][51][52][53][54][55][56][57][58][59][60][61][62][63] Due to the involvement of the conical intersection region in the photodissociation of methylamine, computational methods should be chosen carefully to correctly describe the strong couplings between the electronic states. The ground and rst excited singlet states were studied along four N-H bond dissociation potential energy curves with XMS-PDFT and FMS-PDFT. These four paths correspond to the N-H bond ssions with conformations shown in Fig. 1 are denoted by eclipsed-H3, staggered, 95 and 100 , respectively. FMS-PDFT was tested only for the staggered conformation, but MC-PDFT and XMS-PDFT were tested for all four.

The calculated potential energy curves along the four paths are plotted in Fig. 4. The potential energy curves calculated by both new methods show correct topographies for avoided crossings near the conical intersection seam, both globally and in the zoomed-in regions. However, the distance at which the minimum energy separation occurs is predicted to be shorter by MC-PDFT and XMS-PDFT than by XMS-CASPT2. The N-H bond distances at the minimum energy separation and the corresponding energy separations are listed in Table 4. For all four paths, the PDFT bond distances at the avoided crossing are about 0.056 Å longer than those predicted by XMS-CASPT2. However, Table 4 also shows that the minimum energy separations all agree within 0.08 eV.

Although MC-PDFT does not diagonalize an effective Hamiltonian matrix in the last step, we note that MC-PDFT still gives correct topographies of PESs for the tested four paths of methylamine. Generally speaking, though, MC-PDFT cannot be trusted for regions near conical intersections.

Phenol (C 6 H 5 OH)

The O-H bond dissociation in phenol has been well studied in the past and it can be used as a model system for testing whether a method gives a proper description of potential energy curves for photodissociation. We tested MC-PDFT, XMS-PDFT, FMS-PDFT, and FMS-PDFT/DMRG for the O-H dissociation in phenol with an H-O-C-C dihedral angle of 1 (nearly planar) or 10 .

Fig. 5 shows that the MC-PDFT potential energy curves are qualitatively wrong at both angles, with a double crossing when the dihedral angle is 1 and a lack of avoidance at 10 . The XMS-PDFT method successfully produces avoided crossings near 2.2 Å for both torsion angles with minimum energy separations of 0.04 and 0.28 eV for 1 and 10 , respectively. The corresponding O-H distances are 2.21 and 2.15 Å.

In regions that are far away from the avoided crossings for each dihedral angle, the XMS-PDFT curves agree very well with the MC-PDFT ones. However, we noticed that for this molecule XMS-PDFT presents a noticeable "bump" aer the avoided crossing. This is apparently because the geometry dependence of the offdiagonal elements of the effective Hamiltonian matrix is not consistent enough with the geometry dependence of the diagonal elements. However, the bumps are no greater than 0.07 eV, corresponding to 1.6 kcal mol À1 , which is usually accurate enough for treating electronically excited states. The FMS-PDFT method is tested for O-H dissociation with the H-O-C-C dihedral angle at 10 . Fig. 5(c) shows that FMS-PDFT also succeeds in removing the unphysical double crossing of MC-PDFT. Similar to the issue discussed in regard to the LiH test, the bump near 1.3 Å for both states again results from an interaction between the highest included state and the lowest unincluded state. This analysis is conrmed by Fig. 6, which shows that including the third state in the model space replaces the bump with an avoided crossing. However, a new bump now occurs near 1.5 Å due to the interaction between the third and the uninvolved fourth states. The bump could be removed by involving more states in the SA-CASSCF calculation, but a bump due to the interaction between the highest involved state in the SA-CASSCF calculation and higher states not included in the SA calculation is inevitable (although occasionally one is lucky enough that this only occurs at such a high energy as to be insignicant for practical purposes). This is another case that shows it is sufficient in FMS-PDFT to consider only (N À 1) rotations between adjacent states in an N-state calculation.

Because DMRG can be used to extend CASSCF to large systems with large active spaces, we also implemented and tested FMS-PDFT based on DMRG. This test involves two-state treatment of the phenol molecule with the H-O-C-C dihedral angle at 10 . To verify the accuracy of the DMRG implementation, the active space used in FMS-PDFT/DMRG is the same as that used in FMS-PDFT. Table 5 provides the differences between FMS-PDFT/DMRG and FMS-PDFT for the rotation angles of two reference states and for the energies of the two states for a range of O-H distances. The rotation angles for generating the intermediate basis are different by no more than 0.001 , while the energies for the two states agree within 0.04 meV. The motivation for using DMRG is to study much larger active spaces with MC-PDFT (with state-specic MC-PDFT/DMRG we previously studied 30 active electrons in 30 active orbitals 26 and 34 active electrons in 35 active orbitals 27 ), but the comparison presented here is to show that DMRG agrees well with a conventional solver when the conventional solver is affordable. The good agreement shows that the FMS-PDFT/DMRG method is a promising method for studying the PESs and dynamics of large systems. Altogether, this makes six 3 A 0 states that are considered here (the six 3 A 00 states and the singlet and quintet states are not considered here).

The rst example considered corresponds to an O 2 + O collision with the atom, labeled O3, impinging on the O1 end of the O1O2 diatom. The r(O1O2) distance is 1.208 Å, and the bond angle of the three O atoms is close to linear, a(O2O1O3) ¼ 175 . These two geometric parameters were xed, and the r(O1O3) distance was scanned from 1.0 to 2.5 Å. Fig. 7 shows that there are several avoided crossings as O3 approaches and that the potential curves obtained by XMS-PDFT calculations agree well with those obtained by XMS-CASPT2.

Examination of the conguration interaction coefficients shows that at large r(O1O3), states with congurations corresponding to curves V 2 (blue) and V 4 (purple) leave the six-state model space when r(O1O3) is decreased to $2.5 Å, and two new states arrive. For r(O1O3) < 1.4 Å, these two new states correspond to curves V 2 (blue) and V 3 (green). It is very encouraging that XMS-PDFT agrees well with XMS-CASPT2 even for this rugged landscape with multiple avoided crossings.

Triplet ozone (O 3 )

In the second example, r(O1O2) is again 1.208 Å, and r(O1O3) is xed at 1.4 Å. The scanning parameter is the bond angle, a(O2O1O3), varying from 60 to 180 ; see Fig. 8. This example, like the previous one, also contains several avoided crossings. However, due to the shapes of the curves, it is easier to follow the changes. The avoided crossings clearly show how the ground electronic state (corresponding to curve V 1 ) at 180 correlates to a higher energy state as the bond angle decreases. The avoided crossing of curves V 1 and V 2 is at $160 , that of V 2 and V 3 is at $145 , that of curves V 3 and V 4 is at $115 , that of curves V 4 and V 5 is at $90 , and that of curves V 5 and V 6 is at $85 .

Again, the characteristics of the XMS-PDFT calculations are in strikingly good agreement with those of XMS-CASPT2. There are, however, some minor differences, chief among which is that close to 180 , the XMS-PDFT curves are slightly more rugged than the XMS-CASPT2 curves.

Spiro cation

In this section, we test the 2,2 0 ,6,6 0 -tetrahydro-4H,4 0 H-5,5 0 -spirobi[cyclopenta[c] pyrrole] molecule, which is simply called the spiro cation in this paper. The structure of the spiro cation is shown in Fig. 1 in ref. 17. The spiro cation can be viewed as a mixed-valence compound 19 that is composed of two organic subsystems (one on the le, one on the right) with a hole due to the removal of an electron to make the cation. The hole is partly localized on either the le or the right subsystem, which results in their geometries being slightly different from one another. We denote the geometry when the hole is mainly on the le as geometry A, and that where the hole is mainly on the right as geometry B. Then, as in ref. 17, we dene a reaction path from geometry A to geometry B by using the linear synchronous transit method 64 as

.; 3N atoms (24) where Q g denotes the Cartesian coordinates for N atoms atoms, and x is a parameter varying from À1.5 to 1.5. In particular, when x ¼ À0.5 or x ¼ 0.5, the equilibrium geometry is obtained for the spiro cation. When x ¼ 0, the geometry is an average of geometries A and B, and it can be interpreted as a transition structure for intramolecular charge transfer between the le and right subsystems. For this very difficult test case, the XMS-PDFT curves resemble the MC-PDFT curves, and the XMS-CASPT2 curves resemble the MS-CASPT2 curves, which is the expected result when the intermediate basis is the same as the CASSCF basis (zero rotation angle). The XMS-CASPT2, MC-PDFT, and FMS-PDFT potential energy curves along the path of eqn (24) are plotted in Fig. 9. It can be seen that XMS-CASPT2 and FMS-PDFT both give good results with local minima for the ground state when x ¼ AE0.35 and a local maximum and avoided crossing in the ground state when x ¼ 0. However, MC-PDFT and XMS-PDFT do not show local minima for the ground state near x ¼ AE0.35 or AE0.5, and they show an unphysical dip when x ¼ 0. The great improvement of FMS-PDFT compared with MC-PDFT again shows the value of FMS-PDFT.

Section S3 in the ESI † shows some other mixed-valence cases where XMS-PDFT fails to give the correct topography of PESs.

Conclusions

A general scheme for multi-state MC-PDFT is proposed in this paper. In this scheme, the CASSCF reference states are rotated to a set of intermediate states via a unitary transformation, and an effective Hamiltonian matrix in the intermediate-state basis is constructed using the MC-PDFT method for the diagonal elements and wave function theory for the off-diagonal ones. Two practical methods, XMS-PDFT and VMS-PDFT, for the unitary transformations are proposed in this paper, and they are tested on eight systems exhibiting avoided crossings of two to six states. The XMS-PDFT method uses the transformation proposed by Granovsky for XMC-QDPT; VMS-PDFT chooses the transformation that maximizes the trace of the effective Hamiltonian. We implemented the VMS-PDFT method using a convenient Fourier series expansion, and the resulting method is called FMS-PDFT. Tests are performed on systems with avoided crossings to compare the two new multi-state methods, XMS-PDFT and VMS-PDFT (mainly in the FMS-PDFT version), to state-specic MC-PDFT and the more expensive multi-state method, XMS-CASPT2. We nd that FMS-PDFT, like our earlier but less convenient SI-PDFT, gives reasonable potential energy curves for all test cases examined, and it shows great improvement over MC-PDFT. Similarly, XMS-PDFT gives good results for all systems except the mixed-valence spiro cation. Since XMS-PDFT is less expensive than VMS-PDFT and since it usually gives good results, we expect that both VMS-PDFT and XMS-PDFT will be useful for future work. We also implemented the FMS-PDFT method based on DMRG wave functions as a proposed strategy for calculations with large active spaces. The two new multi-state methods proposed here are preferred to the previous SI-PDFT because they treat the ground state and excited states on an equal footing and they require only a single SA-CASSCF calculation and a single set of orbitals.