Variational Forward-Backward Charge Transfer Analysis Based on Absolutely Localized Molecular Orbitals: Energetics and Molecular Properties
To facilitate the understanding of charge transfer (CT) effects in dative complexes, we propose a variational forward-backward (VFB) approach to decompose the overall CT stabilization energy into contributions from forward and backward donation in the framework of energy decomposition analysis based on absolutely localized molecular orbitals (ALMO-EDA). Such a decomposition is achieved by introducing two additional constrained intermediate states in which only one direction of CT is permitted. These two "one-way" CT states are variationally relaxed such that the associated nuclear forces can be readily obtained. This allows for a facile integration into the previously developed adiabatic EDA scheme so that the molecular property changes arising from 1 forward and back donation can be separately assigned. Using ALMO-EDA augmented by this VFB model, we investigate the energetic, geometric, and vibrational features of complexes composed of CO and main group Lewis acids (BH 3 , BeO/BeCO 3 ), and complexes of the N 2 , CO, and BF isoelectronic series with [Ru(II)(NH 3 ) 5 ] 2+ . We identify that the shift in the stretching frequency of a diatomic π-acidic ligand (XY), such as CO, results from a superposition of the shifts induced by permanent electrostatics and backward CT: permanent electrostatics can cause an either red or blue shift depending on the alignment of the XY dipole in the dative complex, and this effect becomes more pronounced with a more polar XY ligand; the back-donation to the antibonding π orbital of XY always lowers the X−Y bond order and thus red-shifts its stretching frequency, and the strength of this interaction decays rapidly with the intermolecular distance. We also reveal that while σ forward donation contributes significantly to energetic stabilization, it affects the vibrational feature of XY mainly by shortening the intermolecular distance, which enhances both the electrostatic interaction and backward CT but in different rates. The synergistic effect of the forward and backward donations appears to be more significant in the transition metal complexes, where the forward CT plays an essential role in overcoming the strong Pauli repulsion. These findings highlight that the shift in the XY stretching frequency is not a reliable metric for the strength of π back-donation. Overall, the VFB-augmented EDA scheme that we propose and apply in this work provides a useful tool to characterize the role played by each physical component that all together lead to the frequency shift observed.
Introduction<!>Vertical vs. Adiabatic ALMO-EDA<!>Generalized SCF-MI<!>Variational Forward-Backward Analysis of the CT contribution<!>Computational Details<!>Borane Complexes<!>Binding of CO with BeO and BeCO 3<!>Pentaammineruthenium(II)<!>CO and BF<!>Conclusions<!>S2 Nuclear gradient for generalized SCF-MI<!>S4<!>S3 Additional results for BH 3 complexes
<p>The stability of dative bonds in classical Lewis acid-base compounds is controlled by the effect of charge transfer (CT). 1 The widely used concept of donor-acceptor interaction stems from the assumption that charge flows from one fragment to another. In simple adducts like ammonia-borane or transition metal complexes with ligands that are σ-donors only (e.g. NH 3 , H 2 O), the assignment of the donor and acceptor moieties is straightforward, while in many other cases, the donor and acceptor moieties interchange their roles when different orbital interactions are considered, wherein the CT is bi-directional. The relative strength of the forward and backward CT and their cooperativity impose a substantial influence on the physical and chemical properties of these donor-acceptor systems.</p><p>Complexes formed by π-acidic ligands (e.g. CO, N 2 , NO, etc.) with main group or transition metal Lewis acids serve as a prominent class of examples for bi-directional CT.</p><p>The synergistic effect of the σ forward donation (ligand→metal) and the π back-donation (ligand←metal) is a well-established concept (Dewar-Chatt-Duncanson) 2,3 and was extensively studied with various computational and analysis schemes. [4][5][6][7][8][9][10] The most prominent representative of this class of ligands is carbon monoxide (CO), which gives rise to the rich chemistry of a large variety of organometallic compounds ranging from mono-metal complexes (e.g. [Ni(CO) 4 ] 11 ) to small multi-metal clusters (e.g. [Fe 2 (CO) 9 ] 1, 12,13 ). Classical carbonyl compounds exhibit a red shift of 0-300 cm -1 in the CO stretching frequency (ν CO ) relative to that of the free CO molecule (2143 cm -1 ), and the shift can be as large as 681 cm -1 for Na 4 [CrCO 4 ]. 14 On the other hand, some other metal-carbonyl complexes (especially cationic complexes) exhibit a blue shift in ν CO , which were classified as "nonclassical" metal carbonyls. 15 Besides bonding with transition metals, CO also forms adducts with main group Lewis acids where the similar bi-directional CT is observed, including various boron compounds 9,16,17 and intramolecular B/N frustrated Lewis pairs. 18 Most recently, it was shown that CO is able to form octacarbonyl complexes with alkaline earth metals (Ca, Sr, and Ba) in their zero oxidation state, 19 where the ns 2 electrons are promoted to the empty (n-1)d orbitals to facilitate the backbonding to CO, which further indicates the strong π-acidity of CO.</p><p>Dinitrogen (N 2 ), which is isoelectronic to CO, is considered to be a weaker σ-donor due to its more compact electron lone pair and also a weaker π-acceptor because of its larger HOMO-LUMO gap than CO, and consequently the coordination chemistry of N 2 is less rich.</p><p>Nonetheless, since the discovery of the first complex with Ru(II), 20 many transition metal dinitrogen complexes have been synthesized (the interested reader is referred to the published reviews 21,22 ). The complexation of N 2 with transition metals is of crucial importance for artificial nitrogen fixation at ambient conditions, [23][24][25][26][27][28][29][30] which often involves the "activation" of the triple bond, i.e., the back-donation from the metal to the π * antibonding orbital of N 2 .</p><p>In practice, one often measures how activated the N 2 is by measuring the red shift in its stretching frequency. The shift is about 200 cm -1 for the first reported [Ru(II)(NH 3 ) 5 N 2 ] 2+ complex and increases to 400 cm -1 for a recently reported tris(phosphino)silyl osmium complex. 28 Gaining insights into how the chemical environment (such as the metal center and the ligand field) modulate the strength of the backward donation to N 2 will thus play an essential role in understanding the molecular mechanism of nitrogen fixation that facilitates the design of novel and highly efficient catalysts.</p><p>The strength of forwards and backwards direction of CT in the above-mentioned complexes depends on both the π-acidic ligand and the σ-acceptor moiety. To shed light on the nature of donor-acceptor interactions as well as the factors that govern their strength, one can resort to energy decomposition analysis (EDA) schemes [31][32][33] to unravel the effect of charge transfer (along with other binding forces) upon the formation of dative complexes. Originating from the pioneering Kitaura-Morokuma EDA, 34,35 the early variational EDA approaches define CT as the mixing of one fragment's occupied orbitals into virtuals of other fragments. Therefore, starting with molecular orbitals (MOs) optimized on each fragment, one can quantify the energy contribution associated with the CT from one fragment (A) to another (B) by evaluating the change in SCF energy upon the inclusion of the A(occupied)→B (virtual)   relaxations into the variational degrees of freedom. This approach was widely used in early EDA methods, such as the reduced variational space (RVS) 36 and the similar constrained space orbital variation (CSOV) 37 schemes, to identify the forward and backward CT contributions between a pair of fragments. The natural bond orbital (NBO) analysis 38,39 also employs a similar way to define CT between pairs of fragments although very different reference orbitals ("Lewis" orbitals prepared by the NBO procedure that remain strongly orthogonal even between fragments) are employed.</p><p>The more recently developed charge transfer analysis (CTA) based on absolutely localized molecular orbitals (ALMOs) 9 is also able to separate the forward and backward CT contributions for each pair of molecules in a system. Starting from fragment orbitals that are variationally optimized within the supersystem (polarized ALMOs), 40 this approach approximates the CT stabilization energy using the energy lowering associated with a single Roothaan step, i.e., one diagonalization of the supersystem Fock matrix, which is further reformulated with unitary orbital rotations generated from a single-excitation operator (X vo ). 41 The total stabilization energy can then be partitioned into contributions from forward and backward CT between different fragment pairs, which is achieved by evaluating the energetic stabilization associated with each off-diagonal block of X vo in the polarized ALMO basis.</p><p>Moreover, one can perform a singular value decomposition (SVD) on each off-diagonal block of X vo , yielding the complementary occupied-virtual pairs of orbitals (COVPs) that make the most significant contribution to the CT between a specific pair of fragments. 9 While the schemes introduced above can separate the total CT energy into forward and backward contributions and even further into contributions from different pairs of donor and acceptor orbitals, most are unable to characterize the observable effects of CT. The recently developed adiabatic EDA scheme 42 (see Sec. 2.1) represents a systematic step forward to address this gap, which allows one to characterize the effects of CT (and other physical components) on molecular structures and vibrational frequencies. This approach has been employed to investigate observable shifts induced by intermolecular binding, such as the red or blue shifts in vibrational frequencies upon the formation of hydrogen or halogen bonds. [43][44][45][46][47] However, it has not been made generally possible yet to separate the observable effects of forward/backward CT in a manner that is similar to how the ALMO-CTA identifies the A→B and B→A contributions for a pair of fragments A and B, 9 even though such a partition is highly desirable in particular for interpretation purposes. One early exception was the already mentioned CSOV approach, which employed full SCF for one fragment in the field of e.g. frozen orbitals of the other. CSOV was applied for studies of CO bound to metal atoms and clusters. 37,48,49 We note that a decomposition scheme with a similar objective was recently formulated by deriving ALMO-based linear response equations, which, nonetheless, is limited to molecular properties that are only concerned with electronic degrees of freedom, such as static polarizabilities. 50,51 In this work, we extend the original formulation of the adiabatic ALMO-EDA by introducing two additional intermediate potential energy surfaces (PESs). Inspired by the RVS and CSOV approaches, on each of these surface one single direction of CT (either A→B or B→A) is permitted while the other direction remains forbidden. The response of the acceptor fragment to such a one-directional CT, on the other hand, is captured by the intermediate state defined thereof. We obtain these "one-way" CT intermediate states via a special type of self-consistent field (SCF) calculation, whose details are given in Sec. 2.2.</p><p>The variational feature of these two states renders the associated nuclear gradients readily attainable, and thus one can conveniently use them in the context of adiabatic EDA. This opens the door to an in-depth analysis for the effect of forward and backward donations on the energetic, structural, and vibrational features of dative complexes. In Sec. 4, we first validate the results produced by these two "one-way" CT states using two prototypical borane complexes (H 3 N−BH 3 and OC−BH 3 ). We then utilize this approach to investigate the carbonyl complexes of beryllium oxide and carbonate (Sec. 4.2), as well as the complexes composed of a series of π-acidic ligands and the pentaammineruthenium ([Ru(II)(NH 3 ) 5 ] 2+ ) moiety.</p><!><p>The ALMO-EDA scheme 17,52 separates the overall intermolecular interaction energy, ∆E INT , into contributions from frozen interaction (∆E FRZ ), polarization (∆E POL ), and charge trans-fer (∆E CT ):</p><p>where the frozen term can be further decomposed into contributions from permanent electrostatics (∆E ELEC ), Pauli repulsion (∆E PAULI ), and dispersion interaction (∆E DISP ). [52][53][54] Such a decomposition is usually performed at a single given geometry and thus we refer to this approach as the vertical ALMO-EDA thereafter. For details regarding the physical meaning and mathematical definition of each of these terms, we refer the reader to our previous publications. 52,53 Contrasting with the vertical EDA, recently we also proposed an adiabatic formulation of the ALMO-EDA in order to analyze the shifts in molecular properties induced by intermolecular interactions. 42 Instead of decomposing a single-point interaction at a fixed geometry, the geometry of the complex is relaxed at the initial (isolated fragments), intermediate (frozen and polarized), and final (full complex) stages of an ALMO-EDA procedure. As in the vertical version of ALMO-EDA, the frozen state is defined as an antisymmetric product of isolated fragment wavefunctions, and the polarized state is obtained by variationally optimizing the supersystem wavefunction with respect to the orbital rotations on each fragment. 17 These electronic states correspond to distinct PESs, and the geometry relaxation on each of them is facilitated by use of the associated analytical nuclear gradients. Optimization of the isolated fragment and fully relaxed complex geometries can be achieved by employing standard SCF nuclear gradients, and the gradients for the frozen and polarized states have been previously</p><p>derived by some of us. 42 The geometry relaxation on each intermediate surface allows one to obtain information on how each intermolecular interaction component modulates the structure of a complex.</p><p>Moreover, one can perform harmonic frequency calculations at the stationary points evaluated on each surface, thereby obtaining the vibrational frequency shifts induced by each physical component of the interaction. 42</p><!><p>In the original formulation of ALMO-EDA, 17 the polarized yet CT-forbidden state is obtained by using the "SCF for molecular interaction" (SCF-MI) procedure, where one variationally optimizes a fragment-block-diagonal AO-to-MO coefficient matrix with respect to the orbital rotations on each fragment. The variational space of each fragment is thus determined by the span of AO basis functions associated with the atoms that belong to the same fragment.</p><p>The one-particle density matrix (1PDM) can be constructed from ALMOs using</p><p>where C o refers to the MO coefficients for the occupied ALMOs, and σ oo denotes their overlap metric, which is obtained by transforming the AO overlap matrix (S) into the basis formed by these occupied ALMOs:</p><p>The energy functional, E = E[P], can be minimized by solving locally projected SCF equations 40,55,56 or using gradient-based optimization algorithms (with respect to on-fragment orbital rotations). 57 By introducing the concept of fragment variational subspaces, we propose a generalized formulation of SCF-MI. For a given fragment A, instead of using its full AO span (I A ), we define its variational degrees of freedom as G A , where G A is the space spanned by a set of vectors whose expansion coefficients in the AO basis are given by a matrix G A , i.e., G A = span{G A }. The concatenation of subspace vectors for each fragment,</p><p>defines the effective working basis for SCF-MI. The MOs can thus be represented as linear combinations of vectors in G, whose coefficients are denoted as</p><p>. By left-multiplying C G with G, one can retrieve the AO-to-MO coefficient matrix:</p><p>and the 1PDM can thus still be calculated via Eq. ( 2).</p><p>Within generalized SCF-MI, one only requires C G to be fragment-block-diagonal, while the vectors that span the variational subspace of a given fragment (G A , G B , ...) are allowed to be expanded by AO basis functions centered on other fragments. The vectors in G that belong to the same fragment are orthonormalized against each other for convenience, while interfragment orthogonality is usually not enforced. It is evident that the generalized SCF-MI scheme imposes a weaker constraint on the MO coefficient matrix than the original ALMO definition since the AO-to-MO coefficient matrix C need no longer be "absolutely localized", and in fact, if one chooses G A = I A for each fragment, the original AO-based ALMO results will be recovered by generalized SCF-MI. Nevertheless, as C G has a fragmentblock-diagonal structure just like the C matrix in the original scheme, similar equations still hold for generalized SCF-MI once the quantities are properly transformed into the basis defined by G. In Sec. S1 in the Supporting Information (SI), we show the the locally projected SCF equation and the energy gradient with respect to on-fragment orbital rotations for generalized SCF-MI.</p><p>The generalized SCF-MI scheme was originally proposed to allow for a truncated virtual space for each fragment being used in the polarization step of ALMO-EDA, 57 thus providing a well-defined separation between polarization and charge transfer. In addition to that, by modifying the content of G, one can solve a broad spectrum of variational optimization problems with varying degrees of freedom using the SCF-MI procedure. As an extreme example, if one uses the full AO span of the whole system (I) as the variational subspace for each fragment, the full SCF result will be recovered by performing a generalized SCF-MI calculation.</p><!><p>Making use of the flexibility in variational space offered by generalized SCF-MI, we introduce two additional intermediate states into the ALMO-EDA. For a system comprising two fragments, we first denote the converged orbitals (expanded in AOs) at the polarization step as</p><p>, which are absolutely localized on fragments A and B, respectively.</p><p>Note that within generalized SCF-MI, these orbitals can be used to define the variational space for the polarized wavefunction (shown as stage (a) in Fig. 1), and the corresponding MO coefficient matrix C G , according to Eq. ( 4), will be an identity matrix (I). We then construct the virtual space for the full system that is strongly orthogonal to the occupied space, which is spanned by the orthonormalized projected virtual orbitals. The projected virtuals are defined as</p><p>and the orthonormalized vectors, C v,full , are obtained through a canonical orthogonalization 58 of C v,proj . Now we construct two "one-way" CT surfaces: on the first surface, we allow C o,A to be mixed with C v,full and C o,B with C v,B , thus the corresponding G matrix has the form</p><p>] (stage (b) in Fig. 1); on the second surface, we allow C o,B to be mixed with C v,full while C o,A only with C v,A , and the corresponding G matrix is</p><p>] (stage (c) in Fig. 1). From the choice of variational spaces one can infer that A→B donation (but not B→A) is allowed on the first of of these two surfaces, and that the response of the acceptor fragment B, also known as "repolarization", is also captured. The second surface is the reverse: B→A donation (but not A→B) is allowed, with A being repolarized. Finally, as we mentioned above, if one chooses G to be 1), the fully relaxed SCF surface will be recovered.</p><p>We note that the construction of these two "one-way" CT states is inspired by the previ- ously developed RVS 36 and CSOV 37 schemes. One notable difference is that in these previous methods the variational space is constructed from orbitals calculated at the isolated fragment level, while here we start from variationally optimized ALMOs obtained in the polarization step. In addition, in RVS and CSOV only the occupied orbitals on the donor fragment are relaxed while the acceptor fragment orbitals are frozen in their initial shape when considering a given direction of CT; in contrast, in our scheme both fragments are variationally relaxed on these "one-way" CT surfaces. For brevity, in the following we refer to our approach as illustrated in Fig. 1 as the variational forward-backward (VFB) analysis.</p><p>The overall CT stabilization energy in ALMO-EDA is defined as</p><p>where E Tot is the unconstrained, fully relaxed SCF energy of the supersystem and E Pol the energy of the polarized yet CT-forbidden state that is represented by Fig. 1(a). The stabilization effect of each direction of CT, denoted as ∆E CTf (A→B) and ∆E CTb (B→A), can be analogously defined:</p><p>where ). The total charge transfer energy can thus be partitioned into three terms:</p><p>where the higher-order (HO) term captures the non-additive contribution to energetic stabilization arising from the relaxation of electronic structure when both directions of CT are permitted simultaneously. This term is negative and relatively small in all examples investigated in this work (1-3% of the overall strength of ∆E CT for the main group complexes and around 10% for the transition metal complexes). One should note that in the original ALMO-CTA scheme 9 the decomposition of the CT energy also yields the three terms on the right-hand side of Eq. ( 9) but uses a single non-iterative Roothaan-step correction upon the converged ALMO polarized state. Therefore, our scheme based on generalized SCF-MI serves as an alternative approach to decompose the energetic stabilization of CT with the advantage that the energies of the forward and backward CT states are strict upper bound to the full SCF energy. The results of these two approaches will be compared in Sec. 4.</p><p>One desirable feature of the present VFB scheme is that the forces associated with these two "one-way" CT states can be evaluated in the same way as the nuclear gradients for the polarized (AO-based SCF-MI) and fully relaxed (standard SCF) states (see SI Sec. S2 for the mathematical details). Therefore, one can readily augment the adiabatic ALMO-EDA scheme with the PESs of these additional VFB states.</p><!><p>The generalized SCF-MI scheme has been implemented in the released version of Q-Chem 5.0. 59 On top of that, we enabled the energy and force calculations for the VFB "one-way" CT states in a locally developed version so that they can be integrated into the adiabatic ALMO-EDA framework. The original AO-based SCF-MI scheme 40 is utilized in both vertical and adiabatic EDA calculations to separate the polarization and charge transfer contributions.</p><p>To validate the CT energy decomposition results given by our VFB scheme, we compared them against the results of the original perturbative ALMO-CTA. 9 The latter approach was also used to generate the COVPs that help identify the key donor and acceptor orbitals. The COVP orbitals are plotted with an isovalue of 0.1 a.u. and density difference plots with an isovalue of 0.01 a.u.</p><p>All the energy and molecular property calculations were performed using the B3LYP functional [60][61][62] in combination with the def2-TZVPP 63 basis set unless otherwise specified.</p><p>For the 4d transition metals (Ru and Tc), an effective core potential 64 was employed. The B3LYP functional was chosen because it provides adequate accuracy for CT-dominated complexes 65 and also decent agreement with experimental vibrational frequency shifts in all three application examples discussed in this work. In addition, we repeated all the calculations with two other functionals (B97-D 66 and ωB97X-D 67 ) and verified that the qualitative trends given by the VFB analysis can all be reproduced (see Tables S2, S5-S7, and S9-S13 in the SI).</p><p>The optimized structures on different PESs were verified as true minima by examining the lowest harmonic frequency, and all the vibrational frequencies were computed with a finite-difference approach using the analytical nuclear gradients associated with each PES, for which the step size of atomic position displacement was set to be 10 −3 Å. In Sec. 4, we report the frequency shift in the stretching mode of diatomic ligands N 2 , CO, and BF (denoted by XY) relative to that calculated in the isolated (uncoordinated) state:</p><p>A negative value of ∆ν XY corresponds to frequency red shift and a positive value blue shift.</p><p>The adiabatic EDA framework allows one to locate the energy minimum and obtain the associated harmonic frequencies on each individual PES: frozen (Frz), polarized (Pol), forward CT (CTf), backward CT (CTb) and fully relaxed (Tot).</p><p>As the geometry of a complex relaxes when moving from one PES to another, the shift in vibrational frequency calculated in such an adiabatic fashion arises not only from the distinction in constraints applied to different electronic states (i.e. the four states illustrated in Fig. 1) but also from the change in molecular structure. For instance, the inclusion of CT usually shortens the intermolecular distance, which, however, also enhances other effects such as those from electrostatic interactions. In order to estimate the strength of both the electronic and geometric effects of CT, one can perform a constrained geometry optimization in the Pol state with the intermolecular distance fixed at the CTf/CTb minimum-energy distance followed by a frequency calculation. The frequencies obtained thereof are denoted as ν XY (Pol@CTf) and ν XY (Pol@CTb), whose respective differences from the adiabatic CTf and CTb frequencies correspond to the pure electronic effect of forward and backward donation on the vibrational frequency shift:</p><p>where CTx stands for either CTf or CTb.</p><!><p>The VFB decomposition of CT (see Sec. 2.3) yields two new PESs on which only one direction of CT is permitted. As the first step to validate this method, we analyze the H 3 N−BH 3 complex in which the H 3 N→BH 3 forward donation dominates (see the large forward vs. backward ratio for this complex in Table 1). As shown in the left panel of Fig. 2 We then move to the OC−BH 3 adduct that is known to have more involved bi-directional CT as many other carbonyl compounds do: a σ-type forward donation (OC→BH 3 ) from the lone pair on CO into the empty p orbital on boron, and a π-type backbonding (OC←BH 3 ) from the B−H σ bonding orbitals to CO's empty π * level, which are illustrated by the COVPs shown in Fig. 3. The vertical EDA results for this complex, as given in Table 1, reveal its strongly repulsive frozen interaction as well as the substantially favorable polarization and CT contributions. A further decomposition of the CT energy using the scheme introduced in this work suggests that the two directions of CT contribute almost equally in this complex.   The importance of CT for this interaction is further highlighted by the adiabatic ALMO-EDA results (Table 2). On the fully relaxed surface, the obtained C−B and C−O bond distances and the shifts in ν CO are in reasonable agreement with the previous theoretical studies and experimental values. The calculated ν CO on the fully relaxed surface is marginally red-shifted, while that on the Pol surface exhibits a small blue shift. The latter is in line with our previous work 44,71 as well as other studies showing blue shifts in CO stretching frequency in the presence of an electric field along the C→O direction, such as in complexes where CO is bound to a metal cation, 72,73 which is known as molecular Stark effect. This is because the dipole moment of CO increases when the C−O bond is contracted, which is favored by the electrostatic interaction between CO and the positively charged moiety. The blue shift is most prominent on the PES with forward CT only (CTf), which can be attributed to two effects: (i) the forward donation itself (electronic) and (ii) the shortened intermolecular distance r(CB) (geometric) that enhances the electrostatic interaction. The significance of these two effects can be estimated by optimizing the geometry of this complex in the Pol state with the C−B distance constrained at the minimum-energy r(CB) on the CTf surface (1.77 Å) and then evaluating the frequency shift. A blue shift of 105 cm -1 is obtained in this "Pol@CTf" state, which is almost identical to the blue shift on the CTf surface (108 cm -1 ), indicating that the large blue shift caused by forward CT is almost solely a geometric effect, i.e., enhanced electrostatic interaction due to the shortened C−B distance.</p><p>With backward CT only, ν CO is also moderately blue-shifted (24 cm -1 ), for which the geometric and electronic effects can be separated in the same way as described above. With r(CB) fixed at 1.54 Å, the "Pol@CTb" frequency is significantly blue-shifted (+184 cm -1 relative to free CO) than the "CTb@CTb" frequency, indicating that the back-donation, as expected, results in an effective red shift of 160 cm -1 by modifying the electronic structure of the CO moiety. This result, as well as the very small ∆ν CO in the fully relaxed complex, suggests that the shift in ν CO is not always a reliable indicator for the strength of backdonation, as it can be largely compensated by the competing electrostatic force that blueshifts ν CO , which is nicely illustrated by this example. These results are in line with previous discussions in literature, 9,70,72 while our VFB approach is able to directly identify the origin of shifts in r(CB), ∆r(CO), and ∆ν CO .</p><p>In summary, our analysis of the two borane adducts above demonstrates that the VFB decomposition of charge-transfer effects yields qualitatively correct results as one would expect for these systems, validating its use in the applications presented below. It not only serves as an alternative scheme to separate the entire CT stabilization energy into forward and backward contributions, but also directly probes their effects on shifts in observables.</p><!><p>A combined spectroscopic and theoretical study by Frenking et al. 74 investigated the binding of CO with beryllium oxide and carbonate (BeY with Y = O or CO 3 ) as well as the shifts in CO's stretching frequency (ν CO ). These beryllium compounds are strong Lewis acids that are even able to form adducts with noble gases. 75 In their study, carbonyl adducts of two binding modes with either the carbon or oxygen atom of CO interacting with the BeY moiety were isolated and characterized by IR spectroscopy, 74 which are denoted as the κC and κO modes in the following discussion, respectively. It was shown that ν CO is red-shifted for κO carbonyls (−80 cm -1 for CO−BeO and −53 cm -1 for CO−BeCO 3 ) while blue-shifted for the κC isomers (43 cm -1 for OC−BeO and 122 cm -1 for OC−BeCO 3 ). The theoretical investigation demonstrated that the magnitude of interaction energies follows the order OC−BeO > OC−BeCO 3 > CO−BeO > CO−BeCO 3 . Interestingly, the more strongly bound OC−BeO exhibits a significantly smaller blue shift than OC−BeCO 3 . The EDA-NOCV (natural orbital for chemical valence) 33,76 analysis for the two κC complexes revealed that their σ-type forward donations (OC→BeY) are of similar strengths, while the π-type back-donation (OC←BeY) is more pronounced for OC−BeO, which explained both its stronger interaction energy and less blue-shifted ν CO . For the κO complexes, the contribution from CT is slightly weaker and the σ and π donations are of similar magnitude.</p><p>Therefore, instead of backward CT, the authors attributed the red-shifted ν CO in the κO complexes to the "reversed polarization" of CO relative to the κC complexes, which makes the C−O bond longer and less polar.</p><p>We compare the previous findings with the results of our more detailed VFB-EDA analysis including a decomposition of the shifts in ν CO . The ALMO-EDA results at the equilibrium geometries of these four complexes (Table 3) are analyzed first. The comparison between the κC and κO binding modes for BeO reveals that the former is more stable by over 70 kJ/mol, and the difference is evenly distributed over all three EDA terms. The difference in ∆E frz results from the substantially more favorable electrostatic interaction under the κC mode despite its strong Pauli repulsion. Upon polarization, one can observe a reduction of electron density on the distant atom for both binding modes, which is shown in Fig. S2.</p><p>The difference between their CT energies can be rationalized via the COVPs obtained from the perturbative CT analysis (Fig. 4) as well as the CTf/CTb ratio of these two isomers (see Table 3), with both suggesting that the less strong CT in the κO complex stems from the decrease in back-donation. The COVP corresponding to the back-donation (Fig. 4(a)</p><p>and SI Figs. S4(c)-(e)) reveals that under the κC mode the π * acceptor orbital of CO has a better overlap with the π-donor orbital of BeO (i.e. the π * orbital is polarized towards C).</p><p>Furthermore, the CTf/CTb ratios indicate that CT in the κC complex is dominated by back donation (∆E CTf /∆E CTb = 0.5) whereas the κO complex has a slightly stronger forward donation than backward with a CTf/CTb ratio of 1.2.</p><p>The comparison between the BeO and BeCO 3 complexes (see Table 3) indicates that the carbonate analogues bind CO less strongly than the oxides mainly because of their less favorable CT contributions. The CTf/CTb ratios for the BeCO 3 complexes reveal the weaker back-donation in these complexes, which is consistent with the previous EDA-NOCV results. 74 This can be elucidated by the smaller overlap between the π-donor orbital on BeCO 3 with the π * orbital on CO, which is illustrated in Fig. 4. The comparison between  the κC and κO isomers of the carbonate complex, on the other hand, shows the same trend as for the above-discussed BeO complexes. Finally, we note that the higher-order contribution is very small for this set of systems (3% of the overall CT energy for the complexes with BeO and ∼ 1% for those with BeCO 3 ), indicating that the decomposition of CT stabilization energy using the VFB approach yields nearly additive results for these Be complexes.</p><p>The PES scans for OC−BeO and CO−BeO are shown in Fig. 5. Both isomers (κC and κO) are only weakly bound in the frozen state with shallow minima that are located at 2.15 and 2.17 Å, respectively. Polarization strengthens the interaction considerably, which shortens the equilibrium Be−X distances of both isomers significantly. The κC complex has a larger polarization energy, which can be rationalized by the chemically softer lone pair located on the C atom. Interestingly, given the characteristic difference between the lone pairs located on C and O, the energetic contributions from forward donation (from CO/OC to BeO) are of similar magnitude (25-35 kJ/mol) for these two complexes in the range of 1.5-1.8 Å. Nevertheless, we found that the energetic contribution of forward CT (energy lowering relative to the Pol surface) does show a slower decay to zero in the long range within the κC mode, as one would expect based on the more diffuse electron lone pair on the C atom (see Fig. S5 in the SI). The backward donation, on the other hand, is of a greater strength in the κC complex, which is in line with the vertical EDA results (Table 3) and can be explained by the larger amplitude of CO's π * on the C atom. The strong back-donation in the κC complex also results in a reduction of the Be−C distance by 0.1 Å, leading to an intermolecular distance that is only 0.016 Å longer than the full equilibrium distance.</p><p>The most pronounced difference between these two binding modes is observed when comparing their CTf against CTb surfaces for each of them: for CO−BeO, the forward and backward donations yield very similar energetic stabilization relative to the Pol state at all ranges; by contrast, OC−BeO exhibits markedly stronger backward CT than forward around the minimum-energy distance while the back-donation decays more rapidly, resulting in a crossover around 2 Å. Note that the faster decay of backward CT was also observed in the OC−BH 3 example discussed above (Sec. 4.1). Altogether, the CO molecule binds with BeO more strongly under the κC mode mainly due to the more favorable polarization and backward CT contributions. It is noteworthy that the equilibrium Be−X distance on the fully relaxed surface appears to be shorter for the κO complex despite its weaker interaction.</p><p>This, once again, can be rationalized by the more diffuse lone pair (σ-donor orbital) as well as the larger amplitude of the π-acceptor orbital on the C atom, with the former rendering the κC complex more prone to Pauli repulsion and thus less favorable at a short distance, and the latter facilitating a considerable interaction strength even when the intermolecular separation is comparatively large.</p><p>The adiabatic ALMO-EDA results for OC−BeO and CO−BeO are shown in Table 4.</p><p>The carbonyl stretching frequency (ν CO ) is blue-shifted by 102 cm -1 for the κC complex  This is also consistent with the vertical EDA results shown in Table 3. Therefore, the significant blue shift in the fully relaxed OC−BeCO 3 complex (122 cm -1 ) is a consequence of the strong Stark effect, which significantly shortens the C−O bond and increases its stretching frequency, combined with the weak back-donation from BeCO 3 that is inadequate to compensate for the blue-shifting effect.</p><p>The discussion in Sec. 4.1 regarding the shift in ν CO nicely revealed that the enhanced electrostatic interaction upon the shortening of r(C−B) is the main contributor to the frequency shift on the CTf surface by calculating ν CO in the "Pol@CTf" state. Here we employ a similar analysis to investigate the distance dependence of the effects of Frz, Pol, CTf and CTb on ν CO and we pay specific attention to the shifts induced by each EDA component at the full equilibrium X−Be distances (e.g. Frz@Tot) since all these effects are at their maximum strength there. We thus performed constrained geometry optimizations on each intermediate surface with r(X−Be) fixed at varying values and evaluated ν CO at each given X−Be distance for both OC−BeO and CO−BeO. The resulting frequency shifts relative to the free CO stretching frequency, which are denoted as ∆ν CO , are shown in Fig. 6 as a function of r(X−Be). It is clearly revealed that the overall frequency shifts are mainly determined by how the strength of permanent electrostatic interaction, which is encompassed in the frozen term, compares to that of backward CT. Both polarization and forward CT result in small red shifts relative to ν CO on the Frz surface, possibly because of the reduction of Pauli repulsion. At the minimum-energy distance on the Tot surface, the CO−BeO isomer exhibits a minimal blue shift on the frozen surface (+13 cm -1 relative to free CO), small but sizable red shifts induced by Pol (−44 cm -1 relative to Frz) and CTf (−13 cm -1 relative to Pol), and a significant red shift caused by CTb (−85 cm -1 relative to Pol). The sum of these four shifts is −129 cm -1 , suggesting that the contributions from these terms are nearly additive and the higher-order effect (−31 cm -1 ) is small. In contrast, the OC−BeO isomer exhibits a much stronger blue shift on the frozen surface (+199 cm -1 relative to free CO).</p><p>Despite the small red shifts induced by Pol (−21 cm -1 relative to Frz) and CTf (−9 cm -1 relative to Pol) combined with the strong red shift due to CTb (−127 cm -1 relative to Pol), the sum of them is still inadequate to compensate for the strong blue shift induced by frozen interaction, leading to an overall blue-shifted ν CO .</p><p>In summary, our VFB-EDA analysis reveals that the distinct frequency shifts in the κC and κO complexes of beryllium oxide and carbonate result from the competition between the electrostatic interaction and the backward CT from BeO or BeCO 3 to CO. The electrostatic interaction affects the CO stretching frequency through the Stark effect, rendering ν CO strongly blue-shifted for the κC motif while moderately red-shifted for κO. Polarization and forward CT further increase the blue or red shift, mainly by shortening the intermolecular distance, which leads to enhanced Stark effect. The backward CT, which is more pronounced in the complexes with BeO, weakens the C−O bond under both binding modes. It offsets the blue shifts in the κC complexes while further red-shifts ν CO in the κO ones. The results for the κO complexes demonstrate that significant red shift in ν CO is possible even with a weak back-donation if it is aligned with the electrostatic effect. Our results agree with Ref.</p><p>74 in explaining the stronger blue shift in OC−BeCO 3 than in OC−BeO while providing a distinct explanation for the opposite frequency shifts in the κC and κO complexes.</p><!><p>Complexes of the Isoelectronic N 2 ,</p><!><p>The red shift in CO or N 2 's stretching frequency is often used as an indicator for the strength of the π-type metal-to-ligand back-donation, whereas the effect of the forward donation on this vibrational mode is small due to the non-bonding character of the σ-donating orbital (lone pair on the ligand). 72,73,79 Bistoni et al. recently investigated the effects of both σ forward and π backward donations on the C−O bond length and its stretching frequency using charge displacement analysis 80,81 with a large set of classical and non-classical carbonyl complexes. 78 They found that ν CO and r(CO) are in very good correlation with the strength of the π back-donation but not with the strength of the σ forward donation. The blue shift in non-classical carbonyl complexes, on the other hand, stems mainly from electrostatic interaction and polarization due to the positively charged metal moiety. Using the blocklocalized wavefunction (BLW) approach, Mo and co-workers obtained optimized structures and vibrational frequencies of a series of transition metal monocarbonyls with both directions of CT "quenched". 10,77 They revealed that ν CO is uniformly blue-shifted in both neutral and cationic transition metal monocarbonyls when CT is absent, and that CT always results in a red shift in ν CO , which is more pronounced in neutral, classical complexes. The blue shift induced by the transition metal moiety is in line with the Stark effect of CO in an electric field, 72,73 and first arises even without polarization, i.e., on the frozen surface. 44,71 The experimentally observed shift in ν CO is thus a superposition of these effects and the net effect can be small since they induce shifts in opposite directions.</p><p>Boron monofluoride is also isoelectronic to CO and N 2 , and was predicted to be both a stronger σ-donor and π-acceptor compared against CO. [82][83][84] It possesses a lower bond order and like CO, the dipole moment of BF is in the opposite direction of intuition, with the B atom as the negative pole. 85,86 However, BF is only stable under extreme conditions, rendering its coordination compounds difficult to prepare. 87,88 Nonetheless, recent work by Drance et al. reported the synthesis and characterization of a transition metal complex with a terminal coordinating BF ligand and demonstrated BF's σ-donor and π-acceptor properties. 89 Experimental evidence including NMR-, IR-, and Mössbauer-spectroscopy and X-ray crystallography suggests that BF is not only a stronger σ-donor than its isoelectronic counterparts (CO, N 2 ) but also a strong π-acceptor. Differing from the analogous complexes with CO and N 2 that exhibit red-shifted ligand stretching frequencies, the B−F stretching frequency, ν BF , is markedly blue-shifted, which is seemingly contradictory to the assignment of BF as a strong π-acceptor.</p><p>Using the extended adiabatic ALMO-EDA with the VFB states, one is able to characterize and separate the effects of permanent and induced electrostatics, σ forward donation, and π back-donation on the structural and vibrational properties of a given complex. Here we provide a systematic study of octahedral transition metal complexes with the abovementioned three isoelectronic ligands: N 2 , CO, and BF. In order to compare these three ligands directly, we choose a [Ru(II)(NH 3 ) 5 ] 2+ framework because of its simplicity (homogeneous auxiliary ligands (NH 3 ) and low-spin singlet state) and its strong association with π-acidic ligands. Furthermore, the [Ru(II)(NH 3 ) 5 N 2 ] 2+ cation is experimentally accessible and has been well characterized. 20 The vertical ALMO-EDA results at the minimum-energy structure of each complex are summarized in Table 5. Starting with the ammine (NH 3 ) ligand that is a σ-donor only, we find that it exhibits a favorable frozen interaction due to the large dipole moment of NH 3 .</p><p>It also shows a relatively long Ru−N distance (2.2 Å), which gives rise to more favorable electrostatic interaction and weaker Pauli repulsion, respectively. Combined with ∆E Pol (−46 kJ/mol), the total non-CT contribution is significant and constitutes ∼30% of the total interaction energy. The CT term, nonetheless, still makes the largest contribution to binding, and is strongly dominated by forward donation (H 3 N→Ru) as indicated by the large CTf/CTb ratio (10.8).</p><p>The three π-acidic ligands (N 2 , CO, and BF) exhibit drastically different vertical EDA fingerprints than NH 3 . Although both permanent electrostatics and polarization become more favorable in these complexes, they are overlaid by the stronger Pauli repulsion that increases from N 2 to BF (see Table S8 in the SI). Thus, CT becomes the key contributor to these interactions. As indicated by the large magnitude of both ∆E CTf and ∆E CTb values, CT in these complexes is of a typical bi-directional character, and the most significant COVPs (Figs. 7 and 8) clearly demonstrate σ forward donation and π back-donation. The total interaction strength of these complexes is in the order of N 2 < CO < BF, which is also in line with their relative strength of CT. The further decomposition of CT using the VFB approach reveals that N 2 is a much weaker σ-donor as well as a weaker π-acceptor than the more polar CO and BF. The increase in bond polarity (N 2 < CO < BF) in this isoelectronic  The increase in the strength of CT with more polar ligands can also be rationalized with the COVPs shown in Figs. 7 (for N 2 ) and 8 (for CO and BF), which illustrate that more polar ligands are more favored as either σ-donor or π-acceptor. Interestingly, the relative strength of forward donation increases more rapidly than backward donation with the ligand polarity, as indicated by the increasing CTf/CTb ratio from N 2 to BF. It is also noteworthy that although BF is more polar than CO, which is supposed to yield an even better overlap of π * with Ru(II)'s 4d orbitals, the back-donation towards BF is 25 kJ/mol less favorable than that towards CO and the interaction between BF and the Ru(II) moiety is dominated by the σ forward donation. These findings hold for different types of density functionals as shown in Table S9 in the SI.</p><p>The adiabatic EDA results (Table 6) show that the Ru(II) moiety binds the three πacidic ligands (denoted as XY) only loosely on the Pol surface with the minimum-energy Ru−X distances ranging between 2.6-2.7 Å. This is in contrast to the main group beryllium carbonyls discussed in Sec. 4.2 above where short intermolecular distances were already observed on the Pol surface (see Table 4). In the case of N 2 , introducing either the forward or backward CT shortens the Ru−X distance by ∼0.3 Å, indicating the similar σ-donating and π-accepting abilities of this ligand. For CO, the forward CT shortens the Ru−X distance more than backward CT, and this difference is further enlarged in the case of BF, where forward donation shortens the Ru−X distance by over 0.2 Å. This differs from the two main     overcome by the non-CT contributions.</p><p>The comparison of the PES scans for these complexes with respect to the Ru−X distance (Fig. 10) further confirms the variation in the relative strength between the forward and  backward CT. For the N 2 complex (left panel), CTf and CTb are of similar magnitude over the full range of the r(N−Ru), with CTb being marginally more favorable in the short range and a crossover taking place around 1.9 Å; for the CO complex (middle panel), the CTf surface is consistently of a lower energy than that of CTb although the gap between them is small. The BF ligand (right panel), on the other hand, is an even stronger σ-donor but a weaker π-acceptor, with a substantial gap between the CTf and CTb surfaces. Furthermore, from N 2 to BF the frozen surface becomes less favorable, while the stabilization effect due to of polarization (the difference between Frz and Pol) increases. Table 6: Metal-ligand distance [r(M−X)] and bond length of the diatomic ligand [r(X−Y)] (both in Å) evaluated on the Pol, CTf, CTb, and Tot surfaces. In the isolated state, r(N The IR shifts in the stretching frequency of these π-acidic ligands (see Table 7) show an opposite trend to the ordering of the strength of their CT energies (∆E CT in Table 5):</p><p>Table 7: Adiabatic ALMO-EDA results for the shifts in the vibrational frequency of N 2 , CO and BF (∆ν XY in cm -1 ) when associated with the transition metal moieties. a red shift of 150 cm -1 for N 2 , a smaller red shift of 102 cm -1 for CO, and a blue shift of 123 cm -1 for BF. The origin of this seemingly counterintuitive trend can be unraveled by the frequency shifts (∆ν XY ) evaluated on the Pol surface, where the blue shifts increase drastically from N 2 to BF. The blue shift in ν XY for each complex is further increased at the equilibrium structures on the CTf surface, which, as in the OC−BH 3 and OC−BeO cases, is mainly a geometric effect as indicated by the close agreement between Pol@CTf and CTf@CTf frequencies in Table 8. The backward CT, on the other hand, lowers the frequencies relative to their values on the Pol surface for all three ligands, which, as indicated by the three right columns in Table 8, is truly an electronic effect. For the complex with N 2 , the back-donation overpowers the weak electrostatic effect, yielding an overall red-shifted ν N 2 on the CTb surface (−39 cm -1 relative to the free N 2 stretch). The red shift becomes much more pronounced (−150 cm -1 relative to free N 2 ) when both directions of CT are permitted, which can be explained by the combined effect of forward and backward CT in shortening the Ru−N distance. According to the adiabatic EDA results in Table 6, CTf and CTb shorten the Ru−N distance by 0.32 Å and 0.31 Å, respectively, and when combined they shorten the distance by 0.67 Å, indicating that their effects are almost additive in this case. The shortened Ru−N distance then strengthens both the electrostatic effect (blue-shifting) and the back-donation (red-shifting), whereas the latter plays a dominant role in this case.</p><p>The same mechanism applies to both the CO and BF complexes, where the bond shortening effects of CTf and CTb are also nearly additive. However, the relative strength of the blue-shifting electrostatic effect and the red-shifting backward CT, as well as their variation with the Ru−X distance, differs in each complex. The blue shifts on the Pol and CTf surfaces are more significant with more polar ligands, which is in line with the increasingly contracted r(X−Y) from N 2 to BF (see Table 6). The back-bonding to CO appears to exert a strong effect such that the frequency calculated on the CTb surface is red-shifted already</p><p>(5 cm -1 relative to the free ν CO ). The synergy of forward and backward CT further results in a substantial red shift of 102 cm -1 in the fully relaxed complex, suggesting that the strength of backward CT increases more rapidly than the blue-shifting electrostatics with the shortening of the Ru−C distance. The back-bonding to BF, on the other hand, introduces a less pronounced red shift (35 cm -1 relative to ν BF on the Pol surface) such that the frequency evaluated on the CTb surface is still strongly blue-shifted (108 cm -1 relative to the free ν BF ). More interestingly, with both directions of CT permitted, ν BF becomes more blue-shifted relative to that on the CTb surface, which is opposite to the substantial red shifts induced by the synergistic effect of forward and backward CT found in the N 2 and CO complexes, suggesting that the enhancement of the blue-shifting effect overshadows the increase in the strength of back-donation. This is confirmed by the strongly blue-shifted Pol@Tot frequency as shown in Table 9 (314 cm -1 relative to the free ν BF ), which is 142 cm -1 higher than the Pol@CTb frequency. The red shift induced by backward CT, on the other hand, is only increased by 55 cm -1 upon moving from the minimum-energy Ru−B distance on the CTb surface to that on the fully relaxed surface, and therefore it is overpowered by the enhanced blue-shifting electrostatic effect.</p><p>Table 8: Shifts in the stretching frequency of N 2 , CO, and BF (in cm -1 ) when associated with [Ru(NH 3 ) 5 ] 2+ with the Ru−X distance fixed at the optimum values on the CTf ("@CTf") and CTb ("@CTb") surfaces, respectively. Table 9: Shifts in the stretching frequency of N 2 , CO, and BF (in cm -1 ) when associated with [Ru(NH 3 ) 5 ] 2+ evaluated at the fully relaxed minimum-energy Ru−X distance ("@Tot"). We then perform the same set of analyses on NN−[Fe(NH 3 ) 5 ] 2+ , which is the 3d analogue of the Ru(II)−N 2 complex. The 3d electrons are more compact than 4d, resulting in an almost halved magnitude of the ∆E Frz and ∆E CT terms at the equilibrium structure compared to the values of the corresponding Ru(II) complex (see Table 5). Turning to the adiabatic EDA results (Table 6), the Fe−N distance optimized in the Pol state is ∼0.1 Å shorter than the corresponding Ru−N distance, which is consistent with the weaker Pauli repulsion exerted by the 3d orbitals. Nonetheless, once fully relaxed, r(Fe−N) is only ∼0.02</p><p>Å shorter than r(Ru−N) optimized in the same state, indicating the stronger effects of CT in the Ru(II) complex. One should also note that the forward CT has a weaker effect than backward CT on the Fe(II) complex, which is revealed by its CTf/CTb ratio (0.8) as well as the significantly larger Fe−N distance on the CTf surface than on CTb. The decomposition of the shift in ν N2 follows the synergistic mechanism discussed above, whereas the overall red shift (relative to the free ν N2 ) for the Fe(II) complex is slightly more than one half of that for the Ru(II) complex, which is consistent with the relative strength of their ∆E CTb values in Table 5.</p><p>At last we investigate the Tc(I) complex with N 2 , where the metal center is also of 4d 6 configuration but has a different oxidation state. Compared to the Ru(II) complex, the frozen interaction at the equilibrium geometry is over 100 kJ/mol more repulsive (Table 5) due to the more diffuse d-orbitals of Tc(I), while their ∆E Pol and ∆E CTf terms only minimally differ. The backward CT, however, almost triples upon the replacement of Ru(II) with Tc(I), resulting in an overall CT stabilization energy that is over 200 kJ/mol more favorable in the latter. The results of adiabatic EDA further highlight the prominent contribution from backdonation in the Tc(I) complex, such as the significant elongation of the N−N bond (Table 6)</p><p>and the substantial red shift in ν NN (236 cm -1 relative the free N 2 ) that already appears on the CTb surface (Table 7). Despite the overwhelming dominance of CTb, the Tc−N distance on the CTb surface is still over 0.2 Å longer than the fully relaxed equilibrium distance. This demonstrates that even for strong π-donating metals, CTf still plays an important role in</p><!><p>Making use of the flexibility of the generalized SCF-MI scheme, we introduced two new intermediate states in which only one direction of CT is permitted under the ALMO-EDA framework. This allows us to separate a total CT stabilization energy into forward (CTf)</p><p>and backward (CTb) contributions with a residual higher-order term. This new variational forward-backward (VFB) decomposition scheme yields similar results compared to the previously developed ALMO-CTA that employs a perturbative approach. An important difference is that, these two new intermediate states are each variationally optimized such that their energies are strict variational upper bounds to the full SCF energy and the nuclear forces associated with them can thus be readily computed, rendering it possible for one to identify the forward and backward CT contributions to the changes in structural and vibrational properties upon the formation of dative complexes.</p><p>As a proof-of-concept example, we first applied VFB analysis to the OC−BH 3 complex.</p><p>The decomposition of ∆E CT reveals that while CTf and CTb are of similar strength at the minimum-energy distance, the backward donation from BH 3 to CO decays substantially faster in the long range. This steep distance dependence of the strength of back-donation, together with the relatively weak Pauli repulsion due to the electron-deficient nature of BH 3 , elucidates the close agreement between the minimum-energy C−B distances on the CTb and Tot surfaces. The CO stretching frequency (ν CO ) in this system is only minimally shifted relative to that of the free CO, and the adiabatic ALMO-EDA results show that ν CO is strongly blue-shifted on the CTf surface and moderately blue-shifted on the CTb surface, which are seemingly counterintuitive. We identified the significant blue-shifting effect of the electrostatic interaction as it shrinks the CO bond distance to increase the dipole moment of CO, which is further enhanced upon the shortening of the C−B distance driven by CT.</p><p>This blue-shifting electrostatic effect largely cancels out the red-shifting effect of the backdonation.</p><p>We then applied VFB analysis to the carbonyl complexes of BeY (Y = O or CO 3 ), where CO is bound to the BeY moiety under either the κC or κO mode. At their equilibrium structures, the κC complexes are more strongly bound than their respective κO isomers, and the energy differences are rather evenly distributed betwee the Frz, Pol, and CTb terms. The CO stretching frequencies in the κC and κO complexes are blue-and red-shifted, respectively, which agree with the results in Ref. 74. Using the adiabatic ALMO-EDA, we demonstrated that the opposite shifts in ν CO originate from the frozen interaction, which can be attributed  problem can be expressed as</p><p>which has the same form as the standard SCF-MI energy. The four terms on the right-hand side of Eq. ( 1) are nuclear repulsion energy, one-electron energy (kinetic energy and nucleielectron attraction), two-electron energy (Coulomb and exact exchange), and exchangecorrelation energy, respectively. Note that when non-hybrid (pure) functionals are used II reduces to the Coulomb repulsion between electrons.</p><p>The Stoll equation S1 for a given fragment X has the following form in generalized SCF-MI:</p><p>where one "X" in the subscript denotes columns corresponding to fragment X in a matrix and "XX" denotes both rows and columns corresponding to fragment X. P and S are the global 1PDM and the AO overlap matrix, respectively, F is the Fock matrix built from P, and the fragment projector matrix P X in the Stoll equation is defined as</p><p>Finally, X and C G X are the eigenvalues and eigenvectors (fragment MOs in the basis defined by G) to be self-consistently obtained, respectively.</p><p>Besides solving locally projected SCF equations, one can also use gradient-based optimization algorithms to obtain variationally optimized C G . The gradient of energy with S2 respect to orbital rotations in the basis defined by G on a given fragment</p><p>where</p><p>and I G is an identity matrix of G's column dimension. "c.c." refers to the complex of the previous term (transpose in the real case).</p><!><p>Differentiating the generalized SCF-MI energy given by Eq. ( 1) with respect to the positions of nuclei yields</p><p>The first four terms are of the same forms as these components in the nuclear gradient of a standard KS-DFT energy except for the specialized 1PDM. The variable θ X in the fifth term refers to the on-fragment rotations of MOs in the given basis defined by G, which vanishes at the convergence of the generalized SCF-MI problem since by definition one would have</p><p>The second last term in Eq. ( 8) arises from the fact that on each fragment both G X and S3</p><p>The first term on the right-hand side of Eq. ( 9) vanishes because of the stationary condition ∂E/∂C G o,X = 0 for a generalized SCF-MI problem. The second term</p><p>which possesses the same form as the overlap matrix contribution to the standard SCF gradient (the matrix PFP is often called the energy-weighted 1PDM). Now we focus on the last term in Eq. ( 9). Without considering the fragment structure of G, we have</p><p>While in the most general cases the right-hand side of Eq. ( 11) would be non-zero, it does vanish for the solutions to the generalized SCF-MI problems that correspond to the two "one-way" CT surfaces. We first reintroduce the fragmentation: without losing generality,</p><!><p>we assume that fragment A is the donor fragment such that the left index of G A covers all AO basis functions in the system: For our specific problem, the variational degrees of freedom for fragment A are essentially given by the full AO space without any further constraint. Therefore, we have</p><p>Similarly, for fragment B whose variational space is given by all AOs on that fragment, we</p><p>Substituting Eqs. ( 15) and ( 16) into Eqs. ( 12) and ( 13), respectively, we get ∂E/∂G A = 0 and ∂E/∂G B = 0. Therefore, the last term in Eq. 9 also vanishes, suggesting that the overlap matrix contribution to the nuclear gradient can be computed in the same way as a S5 standard SCF problem:</p><p>One should note that Eq. ( 17) only holds for special generalized SCF-MI problems where the variational subspace of each fragment has the same span as the AOs. In cases where this condition is not satisfied, e.g., the definition of each fragment's polarization subspace using fragment electrical response functions (FERFs), S2 one no longer has ∂E/∂G X = 0.</p><p>The variable in the last term on the right-hand side of Eq. ( 8), φ X , refers to the orbital rotations on a given fragment (X) at the polarization stage (the standard AO-based ALMO scheme), which also contributes to the variation of G with the displacement of nuclear positions. However, since we have shown above that E G X SCFMI = 0 for this special problem, the last term in Eq. ( 9) can also be discarded. Taken all together, the nuclear gradients for the two "one-way" CT states are given by</p><p>which is exactly of the same form as the standard SCF nuclear gradients.</p><!><p>Table S1: Decomposition of the frozen interaction energy at the minimum-energy structures using the "quasiclassical" scheme     Table S4: Decomposition of the frozen interaction energy at the minimum-energy structures of the CO−BeY complexes using the "quasiclassical" scheme with the B3LYP functional (energies in kJ/mol).</p>