This thesis presents progress in six areas of quantum photochemistry: 1 a newalgorithm for carrying out efficient semiclassical dynamical calculations using a trajectory surface hopping
ECP-TSH method na c
Trajectory surface hopping (TSH) methods model nonadiabatic transitions as discrete hops between potential energy surfaces These methods integrate electronic coupled equations along classical trajectories to identify hop locations TSH approaches vary in their mechanisms for these transitions, with the TFS and FSTU methods being the most reliable The FSTU method enhances the TFS approach by better addressing frustrated hops that are restricted by energy or momentum conservation.
The ECP-TSH method, created by Parlant and Gislason, permits a trajectory to hop exclusively at points of local maximum coupling, contrasting with the principle that allows hopping whenever the parameter p;; changes In the context of a two-state system, this method effectively utilizes the local maximum coupling to determine trajectory transitions.
A()=|R-4,,(, (7) as a measure of the strength of the coupling Hops are allowed at positions along a trajectory where there is a local maximum of this coupling strength function.
In the ECP-TSH method, the electronic density is reset at all local minima, with a value of unity assigned to the currently occupied electronic state and zeros to all others The hopping probability from state k to k' at each maximum of Q is calculated by integrating along a trajectory between two adjacent local minima Specifically, the probability at the local maximum is influenced by the local minima surrounding it Unlike TFS and FSTU methods, where coherence is maintained throughout the trajectory, the ECP-TSH method destroys coherence after an attempted hop by reinitializing the probability, allowing for a complete coherent passage between local minima before returning to the maximum for the next hop.
Note that Q tends to zero as ¢ > œ after the collision (or photodissociation event) This is interpreted as a local minimum, and so the trajectory can hop at the final
81 local maximum ofQ Similarly /ạ is defined as the start of the trajectory so that a hop can occur at the first local maximum for.
The original ECP-TSH applications by Parlant and Gislason'” used the so-called
The "ants" sampling scheme involves splitting the trajectory at each hopping location into two weighted branches, which are then independently propagated on two potential energy surfaces Each branch has the potential for further branching in the future, enhancing the exploration of the energy landscape.
In their research, 144 utilized a mixed anteater/ants sampling scheme, while Sizun et al employed the Alexander anteater sampling method, which focuses on following a single branch This sampling algorithm incorporates a cutoff parameter, Q, where only maxima exceeding a threshold are identified as potential hopping locations The anteater implementation is applied to enhance the sampling process.
ECP-TSH method, recognizing that for a large enough ensemble of trajectories both the ants and anteater implementations should give the same results.
Although the original ECP-TSH method’? was formulated in the adiabatic representation, there is no reason why it cannot be applied in the diabatic representation.
In this paper, we extend the ECP-TSH method to the diabatic representation, utilizing the same approach as in the adiabatic representation, as outlined in Eq (A9) of Ref 22 Notably, in the two-state scenario, the nonadiabatic coupling vector serves as an effective measure of coupling in both representations The effectiveness of this extension will be evaluated through test cases discussed in Section V.
Self-consistent potential methodS 5< HH HH ng nh Hiệp 82 1 0n S4 2 SCDM HH Hà HT HT Họ Hà Tà nh 84 3 CSDM ố
Decoherent direCfiOT - - cà HH HH TH TH TH nh th 89 3.4.5 Decay-of-mixing time sóc HT TH TH ng HH kh 90 3.5 Three-dimensional test cases and methodology sen neie 92 3.6 Results and ISCUSSIOTI SH TH HT TH TH TT HH 95 3.7 Concluding rermaTKS ch gọn gu nọ TT Hàn ng ng, 101 3.8 AcknowledgmerIs - ch TH TH TH TH HH Hà TH TH nh cư 104 [.ÂU09it 020
In SCDM, the decoherent direction was originally given by!!
$s= x.aoP(Đâ xp + FvipPvip Vlas apP a xe + PyipPvin|| › (23) where dg is a bohr length, P,;,, and dx; are unitless unit vectors in the direction of
P, i, (the local vibrational momentum22) and d x; , respectively, dx, is the magnitude of dr, pid) is the component of P,;, in the direction of d , and K is the decoherentKk› "Kr p vib Kk state The sign in Eq (23) was chosen such that the summation is additive Since both dx, and P,;,, are within the non-rotational subspace, using $ as the decoherent direction conserves total angular momentum There is some ambiguity is separating vibrational and rotational motion, i.e., in the definition of Pv¡p To eliminate this ambiguity, we replace pi) with P-d Kx (1.e., the component of total momentum in the direction of dx; ) This yields
= (3yao[P -â xx ly + Priv Prin Va xeag (Pax lầy + Pirin | (24)
Equations (23) and (24) are for the two-state case; see Ref 11 for a generalization of Eq.
(23) to multiple states in both adiabatic and diabatic representations, and we can generalize Eq (24) in the same way.
The SCDM and CSDM algorithms can be utilized with various models for decay-of-mixing times, including those based on phase decay and the condition that demixing vanishes at low nuclear momentum Previous studies by Fiete and Heller, as well as Turi and Rossky, have explored short-time and perturbative treatments of Gaussian wave packets, offering insights into the physical decoherence function However, it's crucial to note that physical decoherence and algorithmic demixing, while related, are quantitatively distinct Algorithmic demixing refers to the decay of reduced density matrix elements, which is essential for simulating real systems where both electronic and nuclear coordinates are quantum mechanical A single SCP trajectory only has physical nuclear kinetic energy if the electronic state is pure, yet a reduced density matrix often represents a mixture even when the system is in a pure state Therefore, to create an effective semiclassical algorithm based on independent trajectories, we must develop an algorithmic decay of mixing that does not align exactly with quantum mechanical decoherence.
Future research may enhance semiclassical justifications for the decay of mixing rates, yet this study relies on a straightforward approach that meets two key criteria: (1) For low P-8 values, Eq (16) mandates specific outcomes.
To prevent demixing, it is essential that the momentum facilitating the coupling of electronic and nuclear motion is adequate to sustain the necessary energy transfer Additionally, the demixing duration must exceed the shortest electronic time scale relevant to the situation, implying a relationship where Ứ;; is greater than or equal to V.
TK h Two of the simplest possible functions that satisfy these constraints are
In our study, we present equations (27) and (28), which feature unitless parameters C and C” that are equal to or greater than one, alongside a positive energy parameter Eg Equation (27) aligns with the framework established in our previous research, while equation (28) resembles the structure from our original decay-of-mixing paper when C is set to zero We have conducted tests on various forms of these equations in our current work.
Our analysis, conducted with values of C ranging from 1 to 5, demonstrates that the results are largely unaffected by the specific form of 7y or the parameter values, as long as 7; is sufficiently large Appendix B provides comprehensive results using three different values of Eg, further illustrating this insensitivity Additionally, the results show even less sensitivity to changes in C, leading us to adopt Eq (28) with C set to 1 for our final calculations.
All of the decay-of-mixing results given in this paper follow the formalism of Eqs (8)-(22), (24), and (28) and differ only in how the switching probability is calculated.
3.5 THREE-DIMENSIONAL TEST CASES AND METHODOLOGY
We utilize the SCDM, CSDM, CSDM-C, and ECP-TSH methods across five fully-dimensional model systems with varying initial conditions, encompassing a total of twelve test cases, as outlined in Ref 11 Previous work has detailed the model surfaces and the precise quantum mechanical calculations for the MXH27 and YRH2 systems Each model system features two electronic states represented by a diabatic potential energy matrix (PEM), which consists of two diagonal potential energy surfaces and a coupling surface Our test suite examines electronically nonadiabatic atom-diatom collisions, all of which follow a specific form.
A+BC (Et, (29b) where (A, B, C) = (M, H, X) and (Y, R, H) for the MXH and YRH systems, respectively,the asterisk indicates electronic excitation, v and / are the initial vibrational and
93 rotational quantum numbers, and the final internal (i.e., rovibrational) energy of the diatomic fragment is Ej, for reaction products and Ej, in the quenched arrangement./,
The initial conditions of the diatomic molecule are defined by the total energy in electron volts (eV) and its initial rotational state, denoted as (E/eV, j) In all scenarios examined, the molecule starts in its ground vibrational state (v = 0), with the total angular momentum of the system being zero, while electronic angular momentum is disregarded.
The nine MXH cases include the SB, SL, and WL parameterizations of the MXH system, utilizing model atom masses of 6.04695 amu for M, 1.00783 amu for H, and 2.01565 amu for X These cases are analyzed with initial conditions set at (1.10, 0), (1.10, 1), and (1.10, 2) For further details on the MXH parameterizations and initial conditions, refer to Reference 27.
The three YRH cases include the YRH(0.1) parameterization with initial conditions of (1.10, 0), and the YRH(0.2) parameterization with initial conditions of (1.02, 0) and (1.10, 6) The atomic masses for Y, R, and H are 10 amu, 6 amu, and 1.00783 amu, respectively For further details on the YRH parameterizations and initial conditions, please refer to Reference 28.
The objective of this study is to evaluate the effectiveness of semiclassical methods in approximating quantum mechanical results, particularly when quantum oscillations are averaged out, as these oscillations often do not manifest in experimental observables To achieve this, quantum mechanical calculations were conducted at various energies near the nominal scattering energy, and the results were subsequently averaged for analysis.
94 elsewhere!1.27.28) In almost all cases, the values obtained by averaging are very similar to the values obtained at the nominal energy.
The study utilized an adaptive integration algorithm tailored for semiclassical trajectory calculations, employing a Bulirsch-Stoer integrator with polynomial extrapolation, which is modified to avoid stepping over local peaks and minima in electronic probabilities Integration parameters were set to agg = 10^-12 Ey and Amin = 10 a.u., ensuring converged results for the YRH and MXH systems The simulation commenced with the lone atom (Y for YRH and M for MXH) positioned 35 ag from the diatom's center-of-mass for MXH and 20 ao for YRH, concluding when product fragments were at least 30 ao apart, with results remaining consistent even when these distances were increased.
In trajectory-based methods, the final state internal energies, denoted as E¡m or Ein, are calculated without quantization These methods reveal that both the relative translational energy and electronic energy stabilize post-collision The internal energy is then derived by subtracting the final relative translational energy and final electronic energy from the total energy It's important to note that for the scenarios examined, U;,, approaches V; asymptotically, with the final electronic energy represented as either V; or V2.
95 TSH and DM trajectories whereas it is some value between V; and ƒ; in Ehrenfest trajectories.) In the Ehrenfest calculations, the quenching probability was computed by the histogram method.
The semiclassical trajectory calculations and the accurate quantum mechanical results are compared for the following six quantities (i= 1, 2, , 6):
Pp _ the probability of reaction, which is the outcome in Eq (29a)
Pg the probability of quenching, which is the outcome in Eq (29b)
Py the total probability of a nonadiabatic event, which is the sum of Pp and
FR the reactive branching fraction, which is defined as Pg/PN
(Elnt) the average internal energy of the diatomic fragment in Eq (29a)
(Exit) the average internal energy of the diatomic fragment in Eq (29b).
For the three probabilities the error ¢;, for quantity iand test case a (nine cases for
The logarithmically averaged percentage error for MXH and three cases for YRH is detailed in previous studies For the remaining quantities, Fp, (E/,,), and (int), the error is defined as the unsigned relative percentage error.
Vig - OF 1 big = quai — *100 (30) ia
For case z, the average error in probabilities is
In the study of energy fractionation within nuclear coordinates, the average error for the quantities Fp, Eint, and Ef is analyzed, with a focus on the relationship expressed by the equation 96 é, (Prob) a 1a = Eig: (31)1 3, H T Me.
&, (Fract) = Eig - (32) a 1a i=4 These were then averaged over the nine MXH cases to give the MXH percentage error
PE(X;MXH) -— Š Ey (X), (33) a=l and similarly for the three YRH cases to give YRH percentage error
PE(X;YRH)= " #z(*), (34) 12 œ=l0 where X= 1 = “Prob” and X= 2 = “Fract” Finally we averaged the two types of errors and two types of systems to obtain “average” mean unsigned percentage errors:
PE(average) =1 > PE(X, = PE(X,YRH) 2
We report numerical results for five quantities in Tables I and III:
PE(Prob;MXH), PE(Fract;MXH), PE(Prob; YRH), PE(Fract;YRH) and PE(average). Semiclassical calculations were performed in the adiabatic (A), diabatic (D), and
IntrOduCfIOn - cọ HH HH TT Tu hi ngàn 117 PM
CR-EOMCCSTD(T) - Ác HH HH Hà TH Tà HH ngư 119 4.3 Calculations and r€Suẽ{S - ch TH 120 4.4 Discussion and concluding rermaTẽkS ỏc ng ng gi th 123 4.5 Acknowledgrm€rif§ - HH HH gu HH TH TH HH ĐH 125 ;s{oc ¿cv E
The CR-EOMCCSD(T) method enhances the EOMCCSD(T) approach by incorporating non-iterative corrections to EOMCC energies, derived from the MMCC formalism This advancement offers refined expressions for improved accuracy in computational chemistry.
The CR-EOMCCSD(T) method reveals 120 differences between the CC or EOMCC and the exact, full CI energies of the electronic states This approach provides distinct expressions for both ground-state (4 = 0) and excited-state ( > 0) energies, highlighting its effectiveness in accurately determining electronic state energies.
In this study, we evaluate the performance of the basic variant ID of the CR-EOMCCSD(T) theory, which involves the energies from CCSD (u = 0) and EOMCCSD (yz > 0) The numerator and denominator terms used for calculating corrections due to triple excitations, N GRO) and pi), have been previously defined.
We calculated potential energy curves for the ground state (V; ) and the first excited state (V2) of ammonia using both MC-QDPT and CR-EOMCCSD(T) with the 6-
The article discusses a specific basis set, labeled as 311+G(3df3pd), focusing on three hydrogen atoms designated as Ha, Hạ, and He It details the labeling of N-H bond distances as Ra, Rp, and Rc, with the nitrogen atom positioned at the origin and HẠ aligned along the y-axis Additionally, it mentions the angles formed by the N-Hp and N bonds.
Hc directions make with the y-axis are denoted by @pand wc, respectively The out-of- plane angle is denoted by a The coordinates of the hydrogen atoms are:
121 Ay: x=0 y=R, cosa z= RA sina Hp: x=Rpceosa sinứp y =—Rg cosa COS @p (2) z=Rpsinứ
}y =—Rc cosa cos @c z= Rc sinứ
We generated four scans for C>,, geometries by setting the two distances Rp and
The study involved setting the angle wp equal to ac while maintaining Rc at 1.020 A We then collected 22 to 23 values of Ra, ranging from 0.8 to 11 A, for each of the four pairs of wp and a.
The MC-QDPT calculations were performed using the HONDOPLUS-v.4.5 electronic structure package, focusing on a full-valence active space consisting of 7 orbitals and 8 electrons for ammonia In this calculation, one inactive orbital, specifically the 1s core orbital of nitrogen, was frozen to remain doubly occupied across all configuration state functions (CSFs) The MC-QDPT method incorporates both single and double excitations from all active orbitals.
The CR-EOMCCSD(T) calculations and the underlying CCSD and EOMCCSD computations were performed with the routines described in Refs 12 and 20; these
122 routines form part of the Michigan State University suite of coupled-cluster programs that are incorporated into GAMESS.”!
The CR-EOMCCSD(T) method, being a single reference approach, encounters challenges in systems with conical intersections, resulting in varying positions of the reference configuration relative to the excited states across different geometries This leads to instances where the excitation energy is negative Detailed energy values for the lowest two adiabatic states across all 91 geometries in scans 1-4 can be found in the supporting information.
To compare the results, we establish the zero of energy for each method based on the energy obtained from the MC-QDPT method for the ground state of ammonia at its equilibrium geometry Using this reference point, we calculate the mean energy value across all evaluated points.
CR-EOMCCSD(T CR-EOMCCSD(T MC-QDPT MC-QDPT z.Áẹ () vg (3): G +12 Q )
In this study, we analyze the mean unsigned deviation (¢) of the MM-EOM-CC energies from the MC-QDPT energies for both adiabatic surfaces, focusing on the ground-state energy (W1) and the excited-state energy (Vy).
VCR-EOMCCSP( _ yMC-QDPT | + |vzR-?oMccsn) _ yMC~QDPT £e= 2x9 @)
The percentage error is defined as
The MC-QDPT method offers several advantages, such as size consistency and the ability to handle open-shell excited states It ensures stability for both ground and excited states across a broad configuration space and is effective for degenerate and quasi-degenerate systems The method incorporates non-dynamical correlation through CASSCF energies, while dynamical correlation is introduced in the second-order MC-QDPT step A significant feature of this approach is its perturb-then-diagonalize strategy, which provides an advantage over traditional diagonalize-then-perturb methods by concluding with a diagonalization step.
As a result, we anticipate achieving reliable approximations of coupled potential energy surfaces, even in the vicinity of intersections or avoided crossings, and our findings confirm that this expectation is indeed fulfilled.
The MMCC theory illustrates that the energy differences between EOMCC and full CI energies, along with the non-iterative corrections to EOMCC energies, can be represented through the generalized moments of the EOMCC equations In this context, the projections of the EOMCCSD equations on triply excited determinants contribute to the numerator of equation 1 Meanwhile, the denominator of equation 1 serves to renormalize the triples corrections, enabling the CR-EOMCCSD(T) method to enhance the performance beyond that of the standard approach.
The EOMCCSD(T) approach enhances the performance of potential energy surfaces in photodissociation processes, as evidenced by a comparison between MC-QDPT and CR-EOMCCSD(T) Specifically, the results for scans 1-4 are illustrated in Figs 1-4 For planar ammonia (scans 1 and 3), the findings indicate that both CR-EOMCCSD(T) and MC-QDPT yield significant insights into the bond-breaking region.
The potential energy curves for the ƒ and V> methods demonstrate excellent qualitative agreement, with both approaches revealing a conical intersection at an N—H distance of 2.10 Å, as illustrated in Figure 1 An inset plot further details the potential energy curves for N-H distances ranging from 1.8 to 2.3 Å Additionally, in non-planar geometries (scans 2 and 4), both the MC-QDPT and CR-EOMCCSD(T) methods exhibit an avoided crossing in their potential curves, depicted in Figures 2 and 4.
The adiabatic energies for ammonia's ground and first excited states, calculated using multi-reference MC-QDPT and single-reference CR-EOMCCSD(T) methods, show an average agreement of 7%, with values of ¢ = 0.36 eV and E = 4.88 eV The potential energy curves along significant one-dimensional cuts through conical intersections and avoided crossings exhibit similar characteristics Additionally, EOMCCSD results without triples correction, found in the supporting information, demonstrate significantly lower accuracy compared to the CR-EOMCCSD(T) findings.
PHOTOCHEMISTRY OF LIFH AND NAFH VAN DER
Vibrational energies and wave funCtiOTS se 204 6.4 Semiclassical trajectory calculations s- sgk 206 6.5 Results and diSCUSSIOT - Gà HH HT HT TH Hà HH Hà Hà nhe 211 6.5.1 Photoabsorption spectra for LIFTH - sa sec ssssvseseesre 211 6.5.2 Semiclassical trajectory photodissociation calculations
The vibrational energies and wave functions for zero total angular momentum were calculated using the ABCSPECTRA®6 software, as previously detailed In summary, the vibrational wave functions were expressed in a specific basis.
205 where the basis functions |B ) have the form
S = $Š is the mass-scaled translational Jacobi coordinate describing the Li to center-of- mass of HF motion, s = s§ is the mass-scaled internal Jacobi coordinate describing the
HF vibrational motion is characterized by the expansion coefficient cy,g, while the distributed Gaussian translational basis function is denoted as /„ The asymptotic eigenstate rovibrational function of HF is represented by đụ;, and y, signifies the eigenfunction corresponding to the total angular momentum J = 0, which describes the rotational state of HF.
7, and the orbital angular momentum of Li with respect to the center-of-mass of HF /.
The basis function index consists of the indices u, 7, /, and m, where wu represents the vibrational quantum number of the isolated HF diatom, and m denotes the translational basis functions To obtain the expansion coefficients for all bound states | nv), the Hamiltonian matrix is calculated and diagonalized as outlined in Eq (9).
The same basis was used for both electronic states The matrix elements in Eq.
(2) were evaluated as the sum
8 8 where we have set n = 0, and the matrix elements
Mon', à' = (ỉ|8o„ |8) q2) are evaluated numerically using the analytic expressions for 1g,’ given in Eq (8).
Semiclassical trajectory photodissociation calculations were carried out using version 6.8.1 of the NAT computer code7! as described elsewhere.24 A few important issues are discussed here.
Electronic angular momentum is neglected throughout the calculation The vibrational states (with total angular momentum J = 0) associated with the van der
The ground electronic state of the Waals complex M -FH can be defined by a trio of quantum numbers (v1, vp, v3), which represent the motion associated with the HF stretch (r), the M-[HF] stretch (R), and the M-[HF] bend (y), respectively.
[HF] indicates the center of mass of HF.
In the semiclassical ensemble, the initial position and momentum for each trajectory were chosen based on a distribution linked to the J = 0 quantized vibrational state (v,, vp, Vy) Morse fits were derived along the three internal Jacobi coordinates of the system—z, R, and y—near the minimum of the potential energy surface.
Waals complex, 1.e., the potential along each degree of freedom was approximated by
The equation Vc(C) = Delt-expl - fc (C-Ceq)P ~ De + Emin describes the vibrational energy dynamics of a complex, where C represents the modes r, R, and y In this model, the vibrational modes are treated as separable, allowing the initial vibrational energy Eyv¡p to be expressed as the sum of the energies derived from Morse fits, each corresponding to the appropriate quanta within the different modes.
Evip(Vy,Vg.Vy) = Ep (Vp) + ERR) +E, (Vy), (14) where
Eclvc) =hoclc +th-xecbe +1] (15)207 and @c and x, c are the frequency and anharmonicity parameter for the mode C, respectively For C= r or R, the values of wc and x, ¢ may be computed from Dc,
Ac, and the reduced mass “ic of mode C, oc = hp |oP2E , (16)2D
For the angular coordinate (C = 7), an effective reduced mass was defined by
The Morse parameters for the LiFH-H, LiFH-J, and NaFH surface fits are detailed in Table II Each trajectory within the ensemble was assigned the correct vibrational energy across three modes, along with a set of random phases, following the quasiclassical approach for each mode.
After setting the initial geometry Qọ and momentum, the phase point was transitioned into an excited adiabatic electronic state with energy hu If the adiabatic energy gap at Qọ varied from the excitation energy hv by more than a defined tolerance ô, the excitation was deemed invalid, prompting the selection of a new geometry and momentum.
208 generated as described above The tolerance ¢used for all of the runs reported here was 0.01 eV.
Three semiclassical trajectory methods for coupled-states dynamics were employed: Tully’s fewest-switches (TFS) method, the fewest switches with time uncertainty (FSTU) method, and the natural decay of mixing (NDM) method The TFS and FSTU methods are categorized as surface-hopping techniques, where each trajectory is influenced by a single potential energy surface and is subject to instantaneous surface switches based on a fewest-switches algorithm Energy conservation is maintained by adjusting the nuclear momentum in relation to the nonadiabatic coupling vector.
In the TFS method, two types of hopping events are identified: successful hops and frustrated hops A successful surface hop involves a transition from one electronic state to another, while a frustrated hop occurs when an upward surface hop cannot conserve energy due to insufficient nuclear momentum to meet the required change for total energy conservation in relation to the nonadiabatic coupling vector.
This article discusses two approaches for managing frustrated hops in electronic systems The first approach, termed the “+” prescription, involves ignoring 8.89 frustrated hops The second approach, known as the VV prescription, allows for the ignoring or reflecting of approximately 9 frustrated hops based on the slope of the target electronic surface and the direction of the nonadiabatic coupling vector Recent findings have highlighted the effectiveness of the VV prescription in addressing these challenges.
The 209 prescription offers greater accuracy compared to the + prescription when evaluated across various test scenarios The FSTU method enhances the TFS method by allowing some frustrated hops to execute nonlocal jumps, adhering to energy and momentum conservation principles, and is governed by the time-energy uncertainty relation This approach categorizes hopping events into three types: successful local hops, successful nonlocal hops, and frustrated hops The frustrated hops can be analyzed using either the VV or + prescriptions Notably, the FSTU method demonstrates a significant improvement in accuracy over the TFS approach While the focus remains on the FSTU-VV method, discussions on FSTU+, TFS-VV, and TFS+ methods are also included.
The NDM method is an enhanced version of the Ehrenfest self-consistent potential method that integrates decoherence into the electronic motion equations, resulting in trajectories that conclude in a pure electronic state In scenarios with strong coupling between surfaces, NDM trajectories follow an average potential energy surface akin to the Ehrenfest surface As the coupling diminishes, the electronic density matrix transitions to a diagonal form, indicating propagation in a single electronic state The diabatic and adiabatic versions of NDM are discussed in References 75 and 23, respectively.
Of energy, and we evaluated this quantity by taking the zero of energy as the energy of infinitely separated M and HF.
Calculations were performed in both adiabatic and diabatic representations, with the adiabatic surface set comprising V1, V2, and d, while the diabatic surface set includes U1, U2, and U3 It is important to note that surface hopping calculations in both representations utilize the nonadiabatic coupling vector d to assess momentum changes during surface hops The initial conditions were chosen based on adiabatic potential energy surfaces, regardless of the propagation representation When necessary, the transformation of initial conditions from the adiabatic to the diabatic representation for propagation was managed as previously outlined.
Trajectories were propagated until the product fragments dissociated at a distance of at least 15 angstroms (Å), specifically when R or R’ exceeds 15 Å, where R’ represents the F-[MH] distance The probability of reaction (Pp), which indicates the formation of de-excited LiF + H products, and the probability of quenching (PQ), representing the formation of Li + HF products in their ground electronic state, were calculated by counting the trajectories that concluded in each of the two final configurations.