Quantum Algorithm for Calculating Molecular Vibronic Spectra Nicolas P. D. Sawaya * Intel Labs, Santa Clara, California, USA Joonsuk Huh † Department of Chemistry, Sungkyunkwan University, Republic of Korea; SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Republic of Korea (Dated: August 2, 2019) Abstract. We present a quantum algorithm for calculating the vibronic spectrum of a molecule, a useful but classically hard problem in chemistry. We show several advantages over previous quantum approaches: vibrational anharmonicity is naturally included; after measurement, some state infor- mation is preserved for further analysis; and there are potential error-related benefits. Considering four triatomic molecules, we numerically study truncation errors in the harmonic approximation. Further, in order to highlight the fact that our quantum algorithm’s primary advantage over classical algorithms is in simulating anharmonic spectra, we consider the anharmonic vibronic spectrum of sulfur dioxide. In the future, our approach could aid in the design of materials with specific light- harvesting and energy transfer properties, and the general strategy is applicable to other spectral calculations in chemistry and condensed matter physics. TOC Graphic Calculating the absorption spectrum of molecules is a common and important problem in theoretical chemistry, as it aids both the interpretation of experimental spectra and the a priori design of molecules with particular opti- cal properties prior to performing a costly laboratory syn- thesis. Further, in many molecular clusters and systems, absorption and emission spectra of molecules are required for calculating energy transfer rates [1]. The widespread use of mature software that solves the vibronic problem is one indication of its relevance to chemistry [2–5]. Many quantum algorithms have been proposed for practical problems in chemistry, chiefly for solving the fermionic problem of determining the lowest-energy con- figuration of N e electrons, given the presence of a set of clamped atomic nuclei [6–14]. However, for many chem- ical problems of practical interest, solving the ground- state electronic structure problem is insufficient. To cal- culate exact vibronic spectra, for instance, an often com- binatorially scaling classical algorithm must be imple- * [email protected]† [email protected]mented after the electronic structure problem has been solved for many nuclear positions [15–20]. In this work, we propose an efficient quantum algo- rithm for calculating molecular vibronic spectra, within the standard quantum circuit model. A quantum algo- rithm to solve this problem, for implementation on a bo- son sampling machine [20, 21], was previously proposed and demonstrated experimentally [22, 23], but to our knowledge no one has previously developed an algorithm for the universal circuit model of quantum computation, nor (more importantly) one that can efficiently include vibrational anharmonic effects while calculating the full spectrum. We are also aware of unpublished work that studies the connection between quantum phase estima- tion and sampling problems [24]. Other related previous work includes quantum al- gorithms for calculating single Franck-Condon factors [25, 26] or low-lying vibrational states [27], algorithms for simulating vibrational dynamics [26, 28], and an experimental photonics implementation [28] that simu- lated several processes related to molecular vibrations in molecules. Though these four works simulate vibrational effects, they do not address the problem of efficiently solving the full vibronic spectrum despite the presence of an exponential number of relevant vibrational states, which is the focus of this work. Note that in this work we use the term “classical” solely to refer to algorithms that run on classical computers for solving the quantum problem of calculating vibronic spectra; we are not re- ferring to methods where nuclear degrees of freedom are approximated with Newtonian physics. In order to calculate a vibronic spectrum, one needs to consider the transformation between two electronic po- tential energy surfaces (PESs). The hypersurfaces may be substantially anharmonic, making accurate classical calculations beyond a few atoms impossible [29–32]. In order to introduce our approach, we begin by assuming that the two PESs are harmonic (i.e. parabolic along all arXiv:1812.10495v3 [quant-ph] 1 Aug 2019
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Quantum Algorithm for Calculating Molecular Vibronic Spectra
Nicolas P. D. Sawaya∗
Intel Labs, Santa Clara, California, USA
Joonsuk Huh†
Department of Chemistry, Sungkyunkwan University, Republic of Korea;SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Republic of Korea
(Dated: August 2, 2019)
Abstract. We present a quantum algorithm for calculating the vibronic spectrum of a molecule, auseful but classically hard problem in chemistry. We show several advantages over previous quantumapproaches: vibrational anharmonicity is naturally included; after measurement, some state infor-mation is preserved for further analysis; and there are potential error-related benefits. Consideringfour triatomic molecules, we numerically study truncation errors in the harmonic approximation.Further, in order to highlight the fact that our quantum algorithm’s primary advantage over classicalalgorithms is in simulating anharmonic spectra, we consider the anharmonic vibronic spectrum ofsulfur dioxide. In the future, our approach could aid in the design of materials with specific light-harvesting and energy transfer properties, and the general strategy is applicable to other spectralcalculations in chemistry and condensed matter physics.
TOC Graphic
Calculating the absorption spectrum of molecules is acommon and important problem in theoretical chemistry,as it aids both the interpretation of experimental spectraand the a priori design of molecules with particular opti-cal properties prior to performing a costly laboratory syn-thesis. Further, in many molecular clusters and systems,absorption and emission spectra of molecules are requiredfor calculating energy transfer rates [1]. The widespreaduse of mature software that solves the vibronic problemis one indication of its relevance to chemistry [2–5].
Many quantum algorithms have been proposed forpractical problems in chemistry, chiefly for solving thefermionic problem of determining the lowest-energy con-figuration of Ne electrons, given the presence of a set ofclamped atomic nuclei [6–14]. However, for many chem-ical problems of practical interest, solving the ground-state electronic structure problem is insufficient. To cal-culate exact vibronic spectra, for instance, an often com-binatorially scaling classical algorithm must be imple-
mented after the electronic structure problem has beensolved for many nuclear positions [15–20].
In this work, we propose an efficient quantum algo-rithm for calculating molecular vibronic spectra, withinthe standard quantum circuit model. A quantum algo-rithm to solve this problem, for implementation on a bo-son sampling machine [20, 21], was previously proposedand demonstrated experimentally [22, 23], but to ourknowledge no one has previously developed an algorithmfor the universal circuit model of quantum computation,nor (more importantly) one that can efficiently includevibrational anharmonic effects while calculating the fullspectrum. We are also aware of unpublished work thatstudies the connection between quantum phase estima-tion and sampling problems [24].
Other related previous work includes quantum al-gorithms for calculating single Franck-Condon factors[25, 26] or low-lying vibrational states [27], algorithmsfor simulating vibrational dynamics [26, 28], and anexperimental photonics implementation [28] that simu-lated several processes related to molecular vibrations inmolecules. Though these four works simulate vibrationaleffects, they do not address the problem of efficientlysolving the full vibronic spectrum despite the presenceof an exponential number of relevant vibrational states,which is the focus of this work. Note that in this workwe use the term “classical” solely to refer to algorithmsthat run on classical computers for solving the quantumproblem of calculating vibronic spectra; we are not re-ferring to methods where nuclear degrees of freedom areapproximated with Newtonian physics.
In order to calculate a vibronic spectrum, one needs toconsider the transformation between two electronic po-tential energy surfaces (PESs). The hypersurfaces maybe substantially anharmonic, making accurate classicalcalculations beyond a few atoms impossible [29–32]. Inorder to introduce our approach, we begin by assumingthat the two PESs are harmonic (i.e. parabolic along all
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normal coordinates), the relationship between the twoPES being defined by the Duschinsky transformation[33],
~q′ = S~q + ~d (1)
where ~q and ~q′ are the vibrational normal coordinatesfor the initial (e.g. ground) and final (e.g. excited) PES,
respectively, the Duschinsky matrix S is unitary, and ~dis a displacement vector.
A vibronic spectrum calculation consists of determin-ing the Franck-Condon profile (FCP), defined as
FCP (ω) =∑
|fi〉|〈0|fi〉|2δ(ω − ωi), (2)
where ω is the transition energy, |0〉 is the vibrationalvacuum state of the initial PES (i.e. of the initial elec-tronic state), and |fi〉 is the ith eigenstate of the finalPES with energy ωi. In practice, the function is desiredto some precision ∆ω. Fig. 1 gives a schematic of the vi-bronic problem for the (a) one-dimensional and (b) mul-tidimensional case, where each parabola or hypersurfacerepresent an electronic PES.
In the photonics-based vibronic boson sampling (VBS)algorithm [20], a change of basis known as the Doktorovtransformation [34] is used to transform between the twoPESs, as this harmonic transformation is directly imple-mentable in photonic circuit elements.
Instead of this direct basis change approach, our workis based on constructing a Hamiltonian that encodes therelationship between the two PESs. This provides mul-tiple advantages, outlined below.
We denote dimensionless position and momentum op-erators as qsk and psk respectively, where s labels thepotential energy surface (s ∈ A,B in this work) and klabels the vibrational mode. These follow standard defi-nitions qsk = (ask + a†sk)/
√2 and psk = (ask − a†sk)/i
√2,
where a†sk and ask are vibrational creation and anni-hilations operators. The notation ~· denotes standardvectors as well as vectors of operators, such that e.g.~qA = [qA0, ..., qAM ]T .
The purpose of our classical pre-processing procedureis to express the vibrational Hamiltonian for PES B interms of qAk, pAk, by making the following transforma-
tions: ~qA, ~pA → ~QA, ~PA → ~QB , ~PB → ~qB , ~pB,where ~Qs and ~Ps are respectively the mass-weighted po-sition and momentum operators of PES s [19]. The fulltransformations are
~qB = ΩBSΩ−1A ~qA + ΩB~d (3)
~pB = Ω−1B SΩA~pA, (4)
FIG. 1. A schematic of the vibronic problem. (a) The one-dimensional case, corresponding to a diatomic molecule, forwhich the only vibrational degree of freedom is the distancebetween the two atoms. The lower and upper parabolas rep-resent the potential energy surfaces (PESs) of the ground andfirst excited electronic states, respectively. The vibronic spec-trum problem consists of calculating overlaps, called Franck-Condon factors (FCFs), between vibrational wavefunctions,producing a plot of intensity versus energy. In the zero-temperature case shown here, the initial state is |0〉, theground vibrational state of the ground electronic PES. Thethickness of the transition arrows vary because the FCFs forthose transitions differ. d is the displacement vector. (b)The multidimensional analogue, where the normal mode co-ordinates of each PES are used. In the harmonic case, the re-lationship between the two hypersurfaces can be described bya transformation that includes displacement, squeezing, androtation operations. For many molecules, the vibronic prob-lem is computationally hard on a classical computer, partlybecause spectral contributions from exponentially many vi-brational Fock states can be present.
where
Ωs = diag([ωs1, ..., ωsM ])12 (5)
and ωsk are the quantum harmonic oscillator (QHO)frequencies of PES s. A more pedagogical explana-tion as well as an alternate formulation are given inthe Supplemental Information. Parameter δ is oftenused[19, 20, 35], defined δsk = dsk
√ωsk/~.
Finally, the vibrational Hamiltonian of PES B is ex-pressed in a standard form as
HB =1
2
M∑
k
ωBk(q2Bk + p2Bk), (6)
after each qBk and pBk has been constructed as a func-tion of the ladder operators of PES A. Hence the low-level building block of our algorithm is a truncated cre-
3
ation operator,
a†i =
Lmax∑
l=1
√l|l〉〈l − 1| (7)
where l denotes a vibrational energy level and the im-posed cutoff Lmax denotes the maximum level. Mappingsto qubits (i.e. integer-to-bit encodings) are discussed inthe SI and errors are analyzed below.
Though this work primarily considers the harmoniccase in order to introduce our methodology, the largestquantum advantage will arise from modeling anharmoniceffects. In fact, because the harmonic approximation isamenable to clever classical techniques that cannot be ap-plied to anharmonic PESs [19], it is expected that quan-tum advantage would be more easily demonstrated forthe anharmonic problems than in the harmonic ones.
FCPs from anharmonicity are vastly more costly toapproximate than the harmonic case using classicalalgorithms—for calculations that include Duschinsky andanharmonic effects, we are not aware of molecules largerthan six atoms that have been accurately simulated[17, 29–32]. Arbitrary anharmonicity can be straight-forwardly included in our quantum algorithm by addinghigher-order potential energy terms to the unperturbed(e.g. Eq. 6) vibrational Hamiltonian H0:
H = H0 +∑
ijk
kijkqiqjqk + ... (8)
The ease with which one includes anharmonic effectsis an advantage over the VBS algorithm [20, 21].
Now that we have outlined the required classical steps,we describe our quantum algorithm for determining theFranck-Condon profile. Unlike most quantum computa-tional approaches to Hamiltonian simulation [12, 36–38],which aim to find the energy of a particular quantumstate, the purpose of our algorithm is to construct a fullspectrum from many measurements.
As the procedure makes use of the quantum phase es-timation (QPE) algorithm [39–41], we use two quantumregisters. QPE is a quantum algorithm that calculatesthe eigenvalues of a superposition of states, acting on aquantum state as
∑ci|ψi〉⊗ |0〉 →
∑ci|ψi〉⊗ |φi〉, where
φi and |ψi〉 are eigenpairs with respect to an implementedoperator. The first register S stores a representation ofthe vibrational state, and the second register E is usedto read out the energy (strictly speaking, it outputs thephase, from which the energy is trivially obtained).S is initialized to |0〉, the ground state of HA. A sim-
ple but key observation is that |0〉 can be written in theeigenbasis of HB , such that
|0〉 =∑
i
ci|ψi〉 (9)
where |ψi〉 are eigenstates of HB and coefficients ci arenot a priori known.
One then runs QPE using the Hamiltonian HB (i.e.implementing U = e−iτHB for some arbitrary value τ),with register E storing the eigenvalues. Many quantumalgorithms have been developed for Hamiltonian simula-tion [36, 42–49], any of which can be used in conjunctionwith the algorithm’s QPE step. Convincing numericalevidence suggests that Trotterization [36, 42] is likely tobe the most viable option for early quantum devices [48].Computational scaling is briefly discussed in the Supple-mental Information.
We define εi as the eigenenergy of |ψi〉, and εi as itsapproximation, where an arbitrarily high precision canbe achieved by increasing the number of qubits in reg-ister E. Degeneracies in ε will be ubiquitous, and wedefine the subspace of states with approximate energy εjas Dj = |ψj1〉, ..., |ψjKj
〉, where Kj is the degeneracyin εj . Measuring register E yields εj with probability∑k∈Dj
|ck|2. Hence—and this is the key insight—values
εj are outputted with a probability exactly in proportionto the Franck-Condon factors of Eq. 2. The measure-ments then produce a histogram that yields the vibronicspectrum. The procedure is depicted in Fig. 2, where forthe zero-temperature case one may disregard register Iand gate V (β). See the Supplemental Information for astep-by-step outline of the algorithm.
Note that this is a different approach from how QPE isusually used. Normally one attempts to prepare a statethat is as close as possible to a desired eigenstate, whereashere we deliberately begin with a broad mix of eigenstatesthat corresponds to the particular spectrum we wish tocalculate.
We highlight four potential benefits of this algorithmover the VBS algorithm [20, 21]. First, the quantumstate in register S is preserved for further analysis, whilein VBS the final state is destroyed. After measurement,the state stored in S is a superposition of states withenergy εj . From several runs of the circuit, one may es-timate this stored state’s overlap with another quantumstate [50], estimate its expectation value with respect toan arbitrary operator, or calculate the transition energyto another PES (i.e. simulate excited-state absorption),though methods for analyzing this preserved informationare beyond the scope of this work. Our QPE-based ap-proach has a similar benefit over the canonical quantumcircuit method for calculating correlation functions [37],which does not provide this kind of interpretable post-measurement state.
The second potential benefit is that, as stated above,anharmonic effects are easily included in our framework.Third, accurate photon number detection for higher pho-ton counts is a major difficulty in experimental quantumoptics [22, 23]; it may be that a scaled-up universal quan-tum computer is built before quantum optical detectorsimprove satisfactorally, though this is difficult to predict.Fourth, while there are error correction methods for uni-versal quantum computers, we do not know of such meth-ods for boson sampling devices.
Even at room temperature, the optical spectrum of
4
FIG. 2. A quantum circuit schematic of the quantumalgorithm for calculating vibronic spectra. In the zero-temperature case, only registers S and E are used, with gateV (β) ignored. Register S, which encodes the vibrationalstate, is initialized to |0〉 ≡ |0〉⊗Nq , the vibrational groundstate of the ground electronic PES. Running the quantumphase estimation (QPE) algorithm with registers S and Eyields quantum state
∑i ci|ψi〉S |εi〉E , where a key insight
is that |ci|2 are proportional to the Franck-Condon Factors(FCFs) for each eigenstate i of vibrational Hamiltonian HB .QFT−1 denotes the inverse quantum Fourier transform, U isa unitary exponential of HB , and H is the Hadamard gate. Ameasurement on register E then yields some value εj , pro-portional to the measured phase. εj is the energy of thetransition to an arbitrary precision. One then produces ahistogram from many runs of the quantum circuit. Note thatthe quantum state Aj
∑k∈Dj
ck|ψk〉 is preserved in register S
for further analysis. In the finite temperature case, a registerI (encoding the initial state) is added, and the constant-depth
operation V (β) is implemented (a constant depth means thatthe computational cost of the operation is independent of thenumber of modes)). After QPE, one then measures both reg-isters E and I, with the contribution to the histogram beingεj minus the energy of the initial Fock state |nI〉.
a molecule can be substantially different from its zerotemperature spectrum [51], necessitating methods for in-cluding finite temperature effects. These effects can beelegantly included by appending additional steps beforeand after the zero temperature algorithm, following pre-vious work [21, 52]. Briefly, one appends an additionalquantum register, labeled I, with the same size as registerS. An operator called V (β) is applied to state |0〉S |0〉I ,which entangles the E and I registers to produce a ther-mofield double state. The remainder of the algorithmproceeds as before, except that both E and I are mea-sured and the contribution to the histogram is modified.We elaborate on this procedure in the Supplemental In-formation.
The least-studied source of error in our algorithm isdue to an insufficiently large QHO cutoff Lmax. It is espe-cially important to study this source of error, both qual-itatively and quantitatively, because the standard classi-cal algorithms for calculating FCFs [3, 15, 17, 29–32] donot directly simulate the vibrational Hamiltonian in theFock basis of PES A, and hence do not suffer from thistype of truncation error. An analysis of Suzuki-Trottererrors will be dependent on the QHO mapping chosenand is left to future work.
FIG. 3. Theoretically exact (solid) and approximate (dotted)vibronic spectra at zero temperature under the harmonic ap-proximation. The transitions considered are SO−2 →SO2+e−
[53]; H2O(D2O)→H2O+(D2O+)+e− [54]; and NO2(2A1) →NO2(2B2) [55] (see Supplemental Information for explicitphysical parameters). After histogram construction, eachpeak was broadened by a Gaussian of arbitrary width 100cm−1. Inaccuracies in the approximate spectra are due to aninsufficiently large cutoff Lmax when representing the larger-δ vibrational mode, where Lmax is the highest energy levelin the truncated ladder operator used to represent the mode.The approximate spectra are included in order to show themain qualitative effect of the truncation error, namely thatlower energy peaks converge rapidly, while higher energypeaks are blue-shifted. Insight into this type of error is valu-able because such truncation errors are not present in stan-dard classical vibronic algorithms. In the approximate data,the L1 errors and cutoffs Lmax are 0.208, 0.231, 0.228, 0.241and 10, 45, 57, 61 respectively for SO2, H2O, D2O, andNO2.
We chose transitions in four triatomic molecules—sulfur dioxide (SO2
− → SO2) [53], water(H2O→H2O++e−)[54], deuterated water (D2O;D ≡ 2H)[54], and nitrogen dioxide (NO2[2A1] →NO2[2B2])[55]—and simulated their vibronic spectrausing one electronic transition from each (the SI includes
5
FIG. 4. Our quantum algorithm will show the largest perfor-mance advantages over classical computers when simulatinganharmonic systems. Here we give an example of an anhar-monic vibronic simulation (black) for SO2
− → SO2, show-ing that the harmonic approximation (orange) is inaccuratefor this transition. Vertical lines represent the stick spec-tra, which are broadened to an arbitrary 100 cm−1. Usingthe expansion in Eq. 8, the Duschinsky transformation andSO2
− PES from Lee et al. [53], and the anharmonic SO2 PESfrom Smith et al. [56], we included the following third- andfourth-order terms: q1q1q1, q1q1q2, q1q2q2, q1q3q3, q2q2q2,q2q3q3, q1q1q1q1, q1q1q2q2, q1q1q3q3, q2q2q2q2, q2q2q3q3, andq3q3q3q3, where subscripts label the normal mode (see SI foradditional details). Many molecules exhibit substantial an-harmonicity, which is the class of molecules for which ourquantum algorithm will be able calculate spectra that classi-cal algorithms are likely unable to.
additional information, including all physical parametersused). We must note that there is a well-studied conicalintersection near the bottom of PES 2B2 in NO2[57–60]which greatly alters the vibronic properties—in theapproximation used here, this feature is ignored. Thelatter three molecules were chosen explicitly becausethey have unusally high phonon occupation numbers fora vibronic transition, making them good candidates fora study on Lmax requirements.
Fig. 3 shows both the theoretically exact vibronic spec-tra (solid line) and an arbitrarily chosen approximatespectrum (dotted line) for each molecule. These plotsshow the qualitative error behavior: higher-energy peaksare blue-shifted while low-energy peaks converge rapidly.It is useful to consider this error trend when implement-ing the algorithm on a future quantum computer. Sim-ulation details and additional analysis are given in theSI.
Finally, in order to highlight the simulation of anhar-monicity as our quantum algorithm’s primary applica-tion, we consider the zero temperature anharmonic spec-
trum of SO2− → SO2, using the same Duschinsky matrix
as before [53] but an anharmonic PES for SO2 takenfrom Smith et al. [56]. Fig. 4 shows the stark differ-ence between the full anharmonic simulation and the har-monic approximation. We included both third-order andfourth-order Taylor series terms (Eq. 8), which are easilymapped to the quantum computer using the same proce-dure as before (see SI for additional details). This rela-tive failure of the harmonic approximation is not uncom-mon [17, 29–32], indicating a well-defined set of molecules(those with substantially anharmonic PESs) for whichour quantum algorithm would outperform classical com-puters.
We introduced a quantum algorithm for calculatingthe vibronic spectrum of a molecule to arbitrary pre-cision. We noted several advantages over the previ-ously proposed vibronic boson sampling (VBS) algo-rithm. First and perhaps most importantly, anharmoniceffects (whose inclusion is very costly classically but of-ten chemically relevant) can be easily included in ourapproach. Second, measuring the eigenenergy in ouralgorithm leaves the quantum state preserved, allow-ing for further analysis that would not be possible inVBS. Third, there are error-related advantages to ourapproach. Aspects of our algorithm may be extendedto other chemical processes for which nuclear degrees offreedom are difficult to simulate on a classical computer.This work’s general strategy, of calculating the energydistribution outputted from quantum phase estimationto arbitrary precision, may be applied to other spectralproblems in chemistry and condensed matter physics.
SUPPORTING INFORMATION.
Elaboration on vibronic Hamiltonian construction,quantum harmonic oscillator to qubit mappings, thefinite temperature algorithm, computational scaling,molecular data and parameters used, and additional er-ror analysis.
ACKNOWLEDGEMENTS
J.H. acknowledges support by the Basic ScienceResearch Program through the National ResearchFoundation of Korea (NRF) funded by the Min-istry of Education, Science and Technology (NRF-2015R1A6A3A04059773). The authors thank Gian Gi-acomo Guerreschi and Daniel Tabor for helpful sugges-tions on the manuscript.
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Supporting Information: Quantum Algorithm for Calculating
Molecular Vibronic Spectra
Nicolas P. D. Sawaya∗
Intel Labs, Santa Clara, California, USA
Joonsuk Huh†
Department of Chemistry, Sungkyunkwan University, Republic of Korea;
SKKU Advanced Institute of Nanotechnology (SAINT),
Sungkyunkwan University, Republic of Korea
(Dated: August 2, 2019)
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CONTENTS
S1. Hamiltonian construction 2
S2. QHO to qubit mappings 4
S3. Finite Temperature Algorithm 5
S4. Outline of Algorithms 7
S5. Computational Scaling 8
S6. Molecular data 9
S7. Error Analysis 11
References 14
S1. HAMILTONIAN CONSTRUCTION
Here we give a more pedagogical summary of the Hamiltonian construction summarized
in the main text. The procedure involves these three transformations: ~qA, ~pA → ~QA, ~PA→ ~QB, ~PB → ~qB, ~pB. Mass-weighted position and moment operators, ~Qs and ~Ps respec-
tively, are [Huh11]
~Qs = Ω−1s ~qs (S1)
~Ps = Ωs ~ps (S2)
with the M ×M matrix
Ωs = diag([ωs1, ..., ωsM ])12 (S3)
where ωsk are the scalar harmonic oscillator frequencies of normal mode k on PES s.