Statistical Analysis of the Effect of Deuteration on Quantum

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Article

Statistical Analysis of the Effect of Deuteration on Quantum Delocalization in CH

5+

Meredith E Fore, and Anne B. McCoy J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b02685 • Publication Date (Web): 07 May 2019 Downloaded from http://pubs.acs.org on May 13, 2019

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The Journal of Physical Chemistry

Statistical Analysis of the E↵ect of Deuteration on Quantum Delocalization in CH+ 5 Meredith E. Fore and Anne B. McCoy

⇤

Department of Chemistry, University of Washington, Seattle, WA 98195, USA E-mail: [email protected] Phone: 206-543-7464

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Statistical approaches are applied to an investigation of how deuterated forms of + CH+ 5 sample the potential energy surface for this ion. In its ground state, CH5 has

been shown to have amplitude in all of the 120 equivalent minima in the potential as well as in the region of each of the 180 low-energy saddle points that connect these minima. Although deuteration quenches this delocalization, the multidimensional nature of the isomerization pathways makes it difficult to visualize and quantify the delocalization of the ground state wave function. In this work, the localization of the ground state wave function is explored through an analysis of projections of the ground state probability amplitude onto various atom-atom distances. Specifically, analysis of two-dimensional projections reveals that the ground state probability amplitude is localized near structures in which the hydrogen atoms in the ion can be divided into those that make up a CH+ 3 part and those that make up in a H2 part. Analysis of fits of projections of the probability amplitude onto distances between pairs of hydrogen atoms, pairs of deuterium atoms and between hydrogen and deuterium atoms to sums of Gaussian functions, allows us to quantify the increased localization of hydrogen and deuterium atoms in the CH+ 3 and H2 parts of the ion. In addition, the amplitude of the wave function in the regions of the potential that corresponds to the transition state for exchange of hydrogen or deuterium atoms between these two parts of the ion is + found to decrease in the partially deuterated forms of CH+ 5 , and is smallest for CH2 D3

and CH3 D+ 2.

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Introduction Carbonium (CH+ 5 ) is an ion of great interest due to its importance in a broad range of chemical processes including astrochemical reaction networks and combustion chemistry. 1–3 It is also one of the simplest super acids. 4–6 Unlike most molecules where the ground state wave function is localized in a single minimum on the potential surface, the ground state 7 8,9 ab of CH+ 5 is delocalized among the 120 equivalent minima. Quasi-classical, basis set,

initio path integral Monte Carlo, 10,11 and di↵usion Monte Carlo studies 12–15 have all shown that for CH+ 5 in its ground state, the amplitude of the wave function in any of these 120 equivalent minima is comparable to the amplitude at the saddle points that connect these minima. This unusual property of CH+ 5 led to the question of whether or not it could be considered to have a structure, 16 and its structure and bonding continues to be of interest. 17 + The highly delocalized ground state of CH+ 5 also makes CH5 a challenging ion for theoretical

and computational investigations and for the interpretation of the spectra that have been reported. 4,5,8,9,18,19 The multidimentional nature of the isomerization pathways that connect the minima makes it difficult to visualize the ground state wave function and to draw insights about the structure of CH+ 5 from these calculations. Independent of how the structure of CH+ 5 is characterized, studies that employ a variety of computational approaches have all concluded that the equilibrium geometry of this ion is characterized by a central carbon atom bound to five hydrogen atoms, where three of the CH bonds are shorter than the other two. Based on a potential energy surface developed by Jin et al., which was fit to electronic energies that were calculated at the CCSD(T)/aug-ccpVTZ level of theory/basis, 20 the shorter CH bonds are 1.088 and 1.11 ˚ A, while the longer CH bonds are 1.20 ˚ A. In addition, the distance between the hydrogen atoms involved in the longer CH bonds is 0.952 ˚ A. This is shorter than the other HH distances, which range from 1.44 to 2.04 ˚ A. We will refer to the carbon and the three more strongly bound hydrogen atoms as the CH+ 3 moiety, while the remaining two hydrogen atoms comprise the H2 moiety. 21,22 the While the best way to classify the bonding in CH+ 5 is a topic of continued debate,

3



structures and energies of the two low-energy transition state structures that separate these minima are well-described by electronic structure theory. 17,23 Specifically, the lowest energy saddle point corresponds to the Cs saddle point structure shown in Figure 1. The motion that connects the minimum in the potential to this saddle point involves a 60 degree rotation of the CH+ 3 moiety about the methyl rotor axis. The Cs saddle point is calculated to be 35 cm

1

higher in energy than the global minimum based on the global potential of Jin et

al., while the harmonic frequency of the vibration that connects the minimum to this saddle point has a frequency of 200 cm 1 . 20 At a slightly higher energy, the C2v saddle point is at 240 cm

1

above the minimum energy structure. The motion that connects this saddle point to

the minimum energy structure involves a rotation of the bond that connects the carbon atom to H3 about an axis that is perpendicular to the plane that contains the carbon atom and the hydrogen atoms numbered 1, 2 and 3 in Figure 1. As such, motion across this saddle point results in the exchange of a hydrogen atom that is in the CH+ 3 moiety with a hydrogen atom that is in the H2 moiety. The harmonic frequency associated with the vibration that connects the minimum to this transition state is 840 cm 1 . As the harmonic zero-point energy in the isomerization coordinates exceeds the energy of the corresponding transition states it is not surprising that analysis of the ground state probability amplitude is delocalized among the 120 minima and these 180 saddle points that connect the minima on the potential energy surface. 7,10,12–15 Despite this delocalization, the geometries that are sampled retain the pair of longer and three shorter CH bond lengths described above, although which hydrogen atom is involved in which type of CH bond is essentially random. This combination of delocalization of the hydrogen atoms among the five bonding positions and localization of the ion in structures in which the ion can be divided into a CH+ 3 and a H2 moiety is what leads to much of the difficulties in developing approaches that allow us to draw insights about the structure of CH+ 5 from the calculated ground state wave function. This level of delocalization of the ground state wave function presents a significant challenge to the field of vibrational spectroscopy, as it impossible to assign the ro-vibrational

4


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spectrum using models that are based on rigid rotor and harmonic oscillator approximations. The first reported rotationally-resolved spectrum of CH+ 5 was published without assignment, 4 and subsequently, the vibrational features of the spectrum have been qualitatively understood. 19,24 A more recent study by Asvany et al. reported a lower temperature, high resolution spectrum. An analysis of this spectrum using combination di↵erences revealed an underlying pattern of energy levels that could be rationalized using the Hamiltonian that describes rotation in five-dimensions. 5,25,26 This energy level pattern also has been captured 9 by large basis set calculations of CH+ 5 for J  3 of Wang and Carrington, although this

work raised some questions about the symmetry assignments of the states involved in the observed transitions. Both experimental and theoretical studies have shown that partial deuteration of CH+ 5 leads to partial localization of the ground state wave function. 13,18,27–29 While the 120 minima on the potential are equivalent, once zero-point energy is introduced, the zero-point corrected relative energies of the minima will di↵er in partially deuterated isotopologues of CH+ 5 . Specifically, when both hydrogen and deuterium atoms are present, there is increased probability amplitude in structures in which hydrogen atoms occupy the H2 moiety. This differential population of the various isotopomers has been rationalized based on their relative harmonic zero-point energies. 12,29 Earlier studies showed that in CH2 D+ 3 the three deuterium atoms are selectively localized in the CH+ 3 moiety, and the height of the barrier for methyl rotation for structures in which this part of the molecule contained both hydrogen and deuterium atoms exceeded the zero-point energy in this vibration. 12 This information is clearly encoded in projections of the ground state probability amplitude onto the HH distances. The present study focuses on the use of statistical analyses to systematically and quantitatively explore the projections of the ground state probability amplitude of CH+ 5 and its deuterated analogues onto distances between pairs of hydrogen atoms, and to use the results of this analysis to obtain insights into how the ground state of this molecular ion samples the potential surface.

5



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Methods Di↵usion Monte Carlo In this study, the probability distributions that characterize the structure of CH+ 5 and its deuterated analogues are obtained using Di↵usion Monte Carlo (DMC). The details of this approach, and our implementation have been described in detail elsewhere. 12,14 The present implementation is based on the early work of Anderson 30,31 in which the ground state wave function is expressed as an ensemble of equally-weighted localized functions in the twelve Cartesian coordinates that describe the geometry of CH+ 5 , g(x

xj ). In the discussion that

follows, we will refer to these functions as walkers. The DMC algorithm is based on the observation that when a Wick rotation is applied to the time-dependent Schr¨odinger equation, e.g. t is replaced by

(x, ⌧ ) =

X

cn (⌧ )

i¯h⌧ , the solution becomes

n (x)

(1)

n

where

cn (⌧ ) = cn (0) exp [ (En and

n

Eref ) ⌧ ]

(2)

and En are an eigenstate and respective eigenvalue of the Hamiltonian of the system

of interest. In the large ⌧ limit the summation in Eq. 1 will be dominated by c0 (⌧ )

0 (x)

since

c0 (⌧ ) shows the slowest decay among the coefficients in the expansion of the wave function. In the simulation Eref is adjusted so

(x, ⌧ ) has constant amplitude as a function of ⌧ , and

the time-averaged value of Eref is E0 . The above procedure relies on the determination of Eref (⌧ ). Following Anderson, 30 the ensemble of walkers is propagated through discrete time intervals of length

⌧ . In each

of these time steps, each of the coordinates that define the position of each walker, xj,i , is p ⌧ /mi , ¯ shifted by a random value taken from a Gaussian distribution with width i = h 6


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where mi is the mass of the atom being displaced. Following the displacement of the walkers, the potential energy of each walker is compared to Eref and

Pj = exp [ (V (xj (⌧ ))

Eref (⌧ )) ⌧ ]

(3)

is evaluated for each walker. The integer component of Pj is used to determine the number of identical walkers at the position of the jth walker at the beginning of the next time step in the simulation. The fractional component of Pj provides a probability of an additional walker being placed at this geometry. Finally, Eref is obtained by evaluating

Eref (⌧ ) = V (⌧ )

↵

N (⌧ ) N (0) N (0)

(4)

where V (⌧ ) is the average potential energy of the N (⌧ ) walkers at time ⌧ , and ↵ is a simulation parameter. The second term introduces a penalty for fluctuations in N (⌧ ), and using ↵ = 1/(2 ⌧ ) has been shown to work well. 32 Once the system has equilibrated, the time average of Eref provides the zero-point energy of the system, while the ground state wave function is represented by the density of walkers. In the present study, we focus on the analysis of projections of the ground state probability amplitude onto atom-atom distances. This requires the evaluation of

2

(x) as well as the

ground state wave function. This is achieved using the descendant weighting approach in which the value of the wave function at the coordinates of the jth walker is equated to the number of descendants that can be attributed to that walker after the simulation has been propagated for a predetermined number of additional time steps. 33,34

Maximum Log-Likelihood Estimation To obtain insights into structure in CH+ 5 and explore trends with deuteration, one-dimensional projections of the ground state probability amplitude, obtained from the DMC simulations, 7



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are fit to sums of Gaussian functions,  K X (q ck )2 Ak p exp Pf (q) = 2 k2 k ⇡ k=1

(5)

Here q represents the coordinate onto which the probability amplitude is projected, ck is the center of the kth Gaussian function,

k

is the width of this function, while Ak is the integrated

area of the kth Gaussian. We choose to fit Pf (q) to a sum of Gaussian functions since the ground state probability amplitude in the harmonic limit is a Gaussian. In earlier studies, we showed that the projections of the probability amplitude based on a harmonic analysis of the ground state of CH+ 5 onto various atom-atom distances resemble the distribution obtained from DMC. 13 These distributions have been calculated based on a harmonic treatment of the three stationary points shown in Figure 1, and are provided in Figures S1 to S3. Based on a fit of the ground state probability amplitude to Eq. 5, we analyze changes in the positions, widths and contributions of individual Gaussian functions to the fit of Pf , and explore how these parameters are a↵ected by partial deuteration. A challenge arises as the accuracy of the fit will necessarily increase with increased K in Eq. 5, but that does not necessarily indicate that the additional parameters are providing additional information or insights. In statistical inference, the likelihood function

L(✓; q) =

N Y

Pf (qj ; ✓)

(6)

j=1

provides an approach for evaluating the quality of a fitted model based on observed data. The likelihood function is defined as the joint probability distribution of observed data, q, treated as a function of the parameters, ✓, in the fitting function, Pf (q). If the fitting function has a set of 3K parameters and is a function of q, the corresponding likelihood function has a set of N samples q1 ...qN and is a function of the set of parameters {✓}. In the present context the number of samples is determined by the number of walkers in the collected wave

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function after the descendant weighting procedure, while ✓ represents the set of A, c and -parameters in Pf (q) in Eq. 5. The set of parameters that maximizes the likelihood function is said to be the most likely set of parameters for a given model and set of data. Substituting Eq. 5 into Eq. 6, the log of the likelihood for this fitting function becomes  K X (qj ck )2 Ak p exp ln(L(q)) = ln 2 k2 k ⇡ j=1 k=1 N X

!

(7)

The maximum log-likelihood is typically used in a log-likelihood ratio test, which provides a quantitative way to compare models based on how the goodness-of-fit is balanced against the complexity of the model. 35 The test is based on the log-likelihood ratio, which uses the maximum log-likelihoods to quantify how much more likely the data are under similar or closely-related models. The test specifically compares nested models, where the more complicated model can be transformed into the simpler model by removing a certain number of parameters from the fit. In this case, this is achieved by adjusting the value of K in Eq. 7. In this study, we use the log-likelihood ratio to assess various models that can be used to fit projections of the probability amplitude obtained from the ground state wave function for CH+ 5 . It is defined as

ln(⇤(q)) = ln(Lmax (✓(0) ; q))

ln(Lmax (✓(1) ; q))

(8)

where Lmax (✓(0) ; q) is the maximum likelihood of the simpler model to fit the data set, and Lmax (✓(1) ; q) is the maximum likelihood of the more complicated model to fit the data set. The likelihood ratio is denoted by ⇤(q). The likelihood-ratio test statistic is and its probability distribution is asymptotically number of degrees of freedom for the

2

2

2 ln(⇤(q)),

as the sample size grows larger. 35 The

distribution is the di↵erence in number of parameters

in ✓(0) and ✓(1) . For the model used in this study, this di↵erence is three. Traditionally, the

2 ln(⇤(q)) statistic would then be used in a

9


2

goodness-of-fit test to


determine whether the addition of new parameters is statistically warranted. However, the sample sizes in this work, which are determined by the number of walkers, are on the order of 107 . It is well-established that p-values go quickly to zero as sample size, N , grows large, depriving it of real meaning. 36 In this work, we find that ln Lmax is maximized for small values of K. As such, rather than numerically maximizing ln Lmax (K), we evaluate ln Lmax for values of K  10 and identify the value of K for which ln Lmax is largest.

Numerical Details In this study, 28 independent DMC simulations were each initialized with 20 000 walkers all starting with the same structure: the minimum energy structure with the carbon atom at the origin and all Cartesian coordinates multiplied by 1.1. The ensembles of walkers were propagated using the potential energy surface of Jin, Braams, and Bowman. 20 The Cartesian coordinates of the minimum energy and saddle point structures, based on this potential surface, are provided in Tables S1 to S3. The simulations were run for a total of 10 000 time steps with a

⌧ of 10 a.u. To obtain samples of | |2 , we record the walkers

in our DMC simulation every 500 time steps starting at 4000 time steps, and track the number of descendants each walker has over the following 50 time steps. The descendants are therefore collected after 4050 time steps through 9550 time steps. In developing distance distributions, the original N walkers are then replicated so the number of walkers at the geometry of the jth walker equals the number of descendants that walker has. This provides twelve independent samples of the ground state probability amplitude for CH+ 5 for each of the 28 simulations, a total of 336, which are used for subsequent analysis. For comparison, we also calculate the ground state probability amplitudes based on harmonic analyses, which have been performed at the three stationary point structures shown in Figure 1. For this analysis, the required Hessians were evaluated using finite di↵erences based on the same potential surface as was used for the DMC simulations. 20 To obtain the projected ground state probability amplitude, the harmonic ground state probability 10


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amplitude is expressed as a product of twelve one-dimensional Gaussian functions based on the twelve normal modes. In the case of saddle point structures, where one of the frequencies is imaginary, the magnitude of the harmonic frequency is used to define the wave functions for these vibrations. We randomly sample geometries of CH+ 5 , and the value of the probability amplitude at these geometries is compared to a random number. The geometries at which the value of the wave function exceeds the random number are used in the subsequent analysis. This provided 109 sampled geometries, which are projected onto the distances between atoms in the same manner as the distribution of walkers obtained from the DMC simulation. Finally, the statistical analysis, described above, requires determining the values of the parameters in Eq. 5 that maximize the log-likelihood function. Given the large data set of roughly 200 000 configurations of CH+ 5 this analysis becomes computationally challenging as K is increased. Rather than finding the values of the parameters in Eq. 5 that maximize the likelihood function, the values of the parameters in Eq. 5 were determined by fitting the projection of the probability amplitude onto the atom-atom distance of interest, which is represented by q in the previous section, using a non-linear least squares fitting approach. The distribution that is being fit was generated by binning the walkers according to the value of the coordinate that the probability amplitude is being projected onto, using 100 bins with a bin width of 0.022 ˚ A, and the fits were performed using the least sqruares function in SciPy. 37 The use of this modified procedure has been tested for fits of the projection of the ground state probability amplitude for CH+ 5 onto the HH distances (PHH (rHH )) for K = 2, 3 and 4. A comparison of the results obtained using these two fitting approaches are provided in Figures S4 and S5 as well as Tables S4 through S6. While there are di↵erences between the parameters obtained using the two approaches, the method used to obtain the parameters does not a↵ect the analysis of the qualitative trends described in the following section.

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Results and Discussion In this study, we focus our analysis on the projections of the probability amplitude for the ground state of CH+ 5 and its deuterated analogues onto the distances between the hydrogen atoms in the ion. In the discussion that follows, we will use the word “hydrogen” to refer to any isotope of hydrogen, while H and D refer to specific isotopes. We then quantify trends in changes in these projections of the ground state probability amplitude through an analysis of the parameters in fits of these distributions to sets of Gaussian functions, as described in the previous section.

Projections onto Hydrogen-Hydrogen Distances To start our analysis, we project the ground state probability amplitude onto the distances between the hydrogen atoms that make up the ion. The resulting distributions are plotted in Figure 2. In Figure 2(a), we explore trends in the sum of the projections of the probability amplitude onto each of the ten distances between the five hydrogen atoms in the ion, PT . As is seen in these plots, the changes in this distribution upon substituting D for H are small, yet are quantifiable. Analogous distributions have been reported and discussed in some of 12,13 our earlier DMC studies of CH+ and the current distributions agree with the results of 5,

those studies. The small di↵erences among these distributions may be surprising as one expects the p width of a vibrational wave function to scale as 4 kµ where k is the force constant for the oscillator and µ represents the associated reduced mass. While the di↵erences among the PT distributions plotted in Figure 2(a) are small, the expected broadening is reflected by the observation that as the ratio of D-atoms to H-atoms in the ion is increased, the probability distribution dies o↵ more rapidly at longer hydrogen-hydrogen distances. In addition, the bimodality of the distribution becomes more apparent as the number of D-atoms is increased. + Despite these di↵erences, overall the changes between the CH+ 5 distribution and the CD5

12


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distribution are subtle. To aid in the interpretation of these distributions, and their insensitivity to the number of D-atoms in CHn D5

n,

in Figure 2(a) we also show the distances between pairs of hydrogen

atoms in the minimum energy structure of CH+ 5 with black vertical lines. The values of these distances are provided in Table 1. The relative heights of these lines indicate the number of distinct pairs of hydrogen atoms that have a given distance in the minimum energy structure. As can be seen, there is one short distance, which corresponds to the distance between the hydrogen atoms numbered 2 and 3 in Figure 1. This corresponds to the part of the distribution near 0.95 ˚ A in which the distribution is flat in CH+ 5 and shows a distinct ˚ peak in CD+ 5 . In addition, the peak that is centered near 1.8 A corresponds to the distances between the atoms that are numbered 1, 4, and 5 in Figure 1 as well as the distances between those which are numbered 2, 4 and 5 in this figure. We use this bimodality to characterize the peak at smaller hydrogen-hydrogen distances as arising from the distance between the two hydrogen atoms that make up the H2 part of the ion (the H2 moiety) and the peak at longer distances as having contributions from distances between pairs of hydrogen atoms, + one or both of which are in the CH+ 3 part of the ion (the CH3 moiety).

We next turn our attention to projections of the ground state probability amplitude onto the HH, HD and DD distances, which are plotted in Figure 2(b)-(d). In the discussion that follows, we will represent these distributions by PHH (rHH ), PHD (rHD ) and PDD (rDD ) respectively, where

PT (r) = PHH (rHH ) + PHD (rHD ) + PDD (rDD )

(9)

The integrated areas under the curves in these plots reflect the number of hydrogen-hydrogen distances that involve the chosen pair of isotopic forms of hydrogen. For example, in CH4 D+ there are six HH distances and four HD distances. The hydrogen atoms that make up the H2 moiety also have longer CH bond lengths and lower CH vibrational frequencies compared to the CH vibrations involving the hydrogen 13



atoms that make up the CH+ 3 moiety. As a result, when the ion is partially deuterated, the H- and D-atoms preferentially localize in the H2 and CH+ 3 moieties, respectively. This is reflected in the di↵erences among the distributions plotted in Figure 2(b)-(d). These di↵erences can be seen most clearly by comparing the plots in panels (b) and (d) in Figure 2, which provide the PHH (rHH ) and PDD (rDD ) for the various isotopologues of CH+ 5 . In Figure 2(d), only the distribution for CD+ 5 (shown with the red solid line) has significant amplitude near 1 ˚ A, while all of the PHH distributions plotted in Figure 2(b) have amplitude near 1 ˚ A. In the case of CH2 D+ 3 the PHH distribution, which is plotted with the green solid line, is localized near 1 ˚ A, with little amplitude in above 1.8 ˚ A. Finally, all of the PHD distributions, shown in Figure 2(c), are localized between 1.5 and 2.3 ˚ A, with peaks at 1.9 ˚ A. These results are consistent with the hydrogen atoms being preferentially localized in the H2 moiety.

Correlations Among Hydrogen-Hydrogen Distances Before applying statistical approaches to analyze the distributions plotted in Figure 2, it is interesting to consider the correlations among the pairs of HH, HD and DD distances in isotopologues of CH+ 5 . In Figure 3, we plot projections of the ground state probability amplitude onto various pairs of HH, HD and DD distances. To distinguish the various atoms we use 0 and

00

superscripts to denote di↵erent atoms of the same type (H or D) in the ion.

The resulting projected probability amplitudes are plotted on a log10 scale to aid in the visualization of the features. In Figure 3(a), we plot the projections of the probability amplitude for CH+ 5 onto two sets of HH distances that share a common hydrogen atom. In CH+ 5 the hydrogen atoms are all equivalent, and the plotted projection of the probability amplitude reflects this symmetry. It has a peak near 1.8 ˚ A, which corresponds to the larger peak in Figure 2(a). There are also two shoulders, which correspond to geometries where one HH distance is roughly 1 ˚ A while the other HH distance is closer to 1.8 ˚ A. These shoulders correspond to configurations of CH+ 5 in which one of the pairs of hydrogen atoms makes up the H2 moiety, while the third 14


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hydrogen atom is in the CH+ 3 moiety. The structure of the distribution shown in Figure 3(a) provides evidence that while the ion is highly fluxional, on average, it is confined to geometries in which one can consider the ion as being composed of H2 and CH+ 3 moieties. This is further illustrated by comparing the projection of the ground state probability amplitude for CH+ 5 , shown in Figure 3(a), to the corresponding projections of the probability amplitude based on a harmonic analysis of the three structures shown in Figure 1. Through this comparison, we find the anharmonic distribution most closely resembles the harmonic distributions obtained using the equilibrium and the Cs saddle point structures, which are provided in Figure S6 and S7. The di↵erences between the ground state probability amplitude obtained from the DMC wave function and from a harmonic analysis based on the C2v saddle point structure show larger di↵erences. For example, the two intense features in the distribution based on the C2v saddle point structure, plotted in Figure S8, which correspond to one HH distance being approximately 1.2 ˚ A while the other HH distance is approximately 1.8 ˚ A, are not as pronounced in the corresponding distribution that is obtained from the DMC ground state wave function. Additionally, only the projection of the ground state probability amplitude based on a harmonic analysis performed at the C2v saddle point structure shows a peak when both pairs of HH distances are roughly 1.2 ˚ A. This feature is not found in the projections of the ground state probability amplitudes shown in Figures 3(a), S6 or S7. The remaining panels of Figure 3 provide projections of the ground state probability amplitudes for partially deuterated isotopologues of CH+ 5 . For example, Figure 3(b) shows 0 00 a projection of the ground state probability amplitude for CHD+ 4 onto the HD and D D

distances. Examination of this distribution shows that the amplitude is localized at DD distances that are roughly 1.8 ˚ A, while there are peaks when the HD distances are roughly 1 ˚ A and 1.8 ˚ A. While the distribution has a shoulder at shorter DD distances and longer HD distances, the amplitude in these geometries is much smaller than what would be expected if the probabilities for the single H-atom occupying the H2 or CH+ 3 moieties followed the

15



2:3 ratio that would be expected if the probability for the H-atom to occupy any of the five bonding positions was equal. These results provide additional support for the picture that when there is only one Hatom in the ion that H-atom will be preferentially localized in the H2 moiety. This picture is further supported by the remaining two projections of the probability amplitude shown in Figure 3(c) and (d). In particular in Figure 3(c) the distribution for CH2 D+ 3 is localized in configurations where the HH distance is near 1 ˚ A, and the DD distances are near 1.8 ˚ A, which corresponds to the two H-atoms making up the H2 moiety, while the three D-atoms are in the CH+ 3 moiety. Finally, the projection of the ground state probability amplitude for CH4 D+ , which is plotted in Figure 3(d), is very similar to the one shown for CHD+ 4, albeit with the axes reversed. This observation follows our expectations based on the above discussions. In this case the D-atom is preferentially localized in the CH+ 3 moiety. The above analysis is also consistent with the partitioning of the ground state probability amplitude 12 for partially deuterated CH+ 5 , reported by Johnson and McCoy.

Analyzing the Structure of CH+ 5 Based on Gaussian Fits To further quantify the above observations, we turn our attention to the fits of the distributions plotted in Figure 2 to sums of Gaussian functions, as indicated by Eq. 5. The maximum log-likelihood value was evaluated for each distribution over a range of K in order to identify the K-value that results in the largest value of ln Lmax . In Figure 4 the ln Lmax values obtained for fits of projection of the ground state probability amplitude onto the DD distances in CD+ 5 are plotted as a function of K. The insets are provided to illustrate how the quality of the fit changes with K. In this example, ln Lmax has a maximum at K = 3. Examining the insets, which are provided at a larger scale in Figures S9 to S12, the fit with K = 2 fails to recover the feature near 1 ˚ A, while for K

3 the distribution is nearly exactly

reproduced. Having identified the optimal number of Gaussian functions needed to reproduce the 16


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probability distributions shown in Figure 2, we focus the remainder of the discussion on trends in the centers of the Gaussian functions obtained from these fits. We have also explored trends for the widths, , and the areas, A, of the Gaussian functions, and find the trends in the centers to be the most informative. Briefly, the areas of the Gaussian functions used to fit the distributions shown in Figure 2(a) with K = 3 have very similar relative values. While these values di↵er as a function of the number of hydrogen atoms when we consider the PHH , PHD , PDD distributions, the trends can be explained by the fraction of hydrogen atoms in the H2 moiety, discussed above. To aid in the comparison of the values of the centers of the Gaussian functions used to fit the projections of the ground state probability amplitude for the various isotopologues of CH+ 5 , plotted in Figure 2, in Figure 5 we plot the values of the ck parameters, which are obtained from these fits. The number of points in each column reflects the value of K that maximizes ln Lmax for that distribution. In the discussion that follows we focus on trends that can be extracted from this analysis. The projection of the probability amplitude onto the distance between the hydrogen atoms in CH2 D+ 3 is the only distribution of HH distances that requires more than two Gaussian functions to fit the distribution. In this case ln Lmax is largest when K = 7. This distribution is unique for its long, non-Gaussian tail, which is difficult to reproduce with a smaller value of K. As noted above and in previous studies, 12,13 CH2 D+ 3 shows an extent of localization that is larger than is found in the other isotopologues of CH+ 5 . Similarly, the projection of the ground state probability amplitude for CH3 D+ 2 onto the DD distance required a value of K = 7 to maximize ln Lmax . Focusing on the PT (r) distributions, we note that the values of the centers of the Gaussian functions used in the fit tend to shift to larger values of r as the number of H- and D-atoms become closer to the same value. To further investigate this trend, in Figure 6 we plot the values of the ck parameters, which are obtained in the fits of the PT (r) distributions with K = 3. We also show the values of the ck parameters that are obtained from fits to the

17



projection of the ground state probability amplitudes of CH+ 5 onto the HH distances, where the ground state wave functions have been evaluated using harmonic analyses at each of the stationary points shown in Figure 1. Focusing on the DMC results for mixed isotopologues, we find that the values of the ck parameters are shifted to smaller values of r compared to the corresponding ck parameters that are obtained from fits to the PT distributions for CH+ 5 and CD+ 5 . While the trend can be found for all three ck parameters, is most easily seen for A. When we compare the results obtained the ck parameter that ranges between 1.4 and 1.6 ˚ from fits to the PT distributions, which are evaluated using the ground state wave function obtained from the harmonic analyses at the three stationary points in Figure 1, we find that all three ck values are largest when the analysis is based on the C2v saddle point structure. In addition, the ck values obtained from analyses performed at the equilibrium and Cs saddle point structures are nearly identical. Combining these observations, we conclude that the ground state vibrational wave functions for the partially deuterated isotopologues of CH+ 5 have less amplitude near the C2v + saddle point structure compared to CH+ 5 and CD5 . The above discussion illustrates how

analysis of fits to projections of the ground state probability amplitude onto atom-atom distances allows us to quantify changes in the way that isotopologues of CH+ 5 sample the transition state regions of the potential. Physically, the loss of amplitude near the C2v saddle point due to partial deuteration reflects the closure of the pathway by which hydrogen atoms + + can move between the H2 and CH+ 3 moieties which make up CH5 or CD5 .

Conclusion In this paper, we have presented an analysis of the vibrational ground state of CH+ 5 and its deuterated analogues. Trends in the localization of H- and D-atoms in partially deuterated isotopologues of CH+ 5 have been presented. These trends have been quantified by analyzing the statistical analysis of fits of projections of the ground state probability amplitude onto

18


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distances between hydrogen atoms to several Gaussian functions. Analyses of projections of the ground state probability amplitude onto distances between two pairs of hydrogen atoms allowed us to further explore how the ground state of CH+ 5 explores the potential energy surface. In particular, the analysis of these distributions reinforces the idea that despite being a highly fluxional molecule, CH+ 5 its structure can be characterized in terms of an H2 moiety and CH+ 3 moiety. Finally, we use these distribuitons to show that the exchange of hydrogen atoms between these two moieties is quenched upon partial deuteration of CH+ 5. This work serves as an example of how statistical analysis of projections of the ground state probability amplitude can be used to generate insights that would have been difficult to obtain from visual inspection of these distributions.

19



Acknowledgement A.B.M acknowledges support from the Chemistry Division of the National Science Foundation (CHE-1619660). MEF acknowledges support through a Graduate Research Fellowship from the National Science Foundation (DGE-1256082 and DGE-1762114). Parts of this work were performed using the Ilahie cluster, which was purchased using funds from a MRI grant from the National Science Foundation (CHE-1624430).

Supporting Information Available Cartesian coordinates of stationary points on the potential; Parameters obtained from fits of PT (r) for CH+ 5 where

2

is minimized and ln Lmax is maximized with K = 2, 3 and 4;

Parameters from the Gaussian fits used in the analysis plotted in Figures 5 and 6; Comparison of the PT distributions obtained from the DMC ground state wave function for CH+ 5 and the results of harmonic analyses at the three low-energy stationary points on the potential; Comparison fits to the PT distributions for CH+ 5 obtained by two fitting approaches; Twodimensional projections of the ground state probability amplitude for CH+ 5 , obtained from harmonic analyses at the stationary points on the potential, onto distances between hydrogen atoms; Larger version of the insets in Figure 4.

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(10) Marx, D.; Parrinello, M. Structural Quantum E↵ects and Three-Center Two-Electron Bonding in CH+ 5 . Nature 1995, 375, 216–218. (11) Kumar P, P.; Marx, D. Understanding Hydrogen Scramling and Infrared Spectrum of Bare CH+ 5 Based on Ab Initio Siumulations. Phys. Chem. Chem. Phys. 2006, 8, 573–586. (12) Johnson, L. M.; McCoy, A. B. Evolution of Structure in CH+ 5 and Its Deuterated Analogs. J. Phys. Chem. A 2006, 110, 8213–8220. (13) McCoy, A. B.; Braams, B. J.; Brown, A.; Huang, X.; Jin, Z.; Bowman, J. M. Ab Initio Di↵usion Monte Carlo Calculations of the Quantum Behavior of CH+ 5 in Full Dimensionality. J. Phys. Chem. A 2004, 108, 4991–4994. (14) McCoy, A. B. Di↵usion Monte Carlo Approaches for Investigating the Structure and Vibrational Spectra of Fluxional Systems. Int. Rev. Phys. Chem. 2006, 25, 77–107. (15) Thompson, K. C.; Crittenden, D. L.; Jordan, M. J. T. CH+ 5 : Chemistry’s Chamelen Unmasked. J. Am. Chem. Soc. 2005, 127, 4954–4958. (16) Schreiner, P. R. Does CH+ 5 Have (a) “Structure?” A Tough Test for Experiment and Theory. Angewandte Chemie, International Edition 2000, 39, 3239–3241. (17) Schreiner, P. R.; Kim, S. J.; Schaefer, I., H. F.; Schleyer, P. v. R. M. CH+ 5 : The NeverEnding Story or the Final Word? J. Chem. Phys. 1993, 99, 3716–3720. (18) Marx, D.; Parrinello, M. Molecular Spectroscopy: CH+ 5 : The Cheshire Cat Smiles. Science 1999, 284, 59–61. (19) Huang, X.; McCoy, A. B.; Bowman, J. M.; Johnson, L. M.; Savage, C.; Dong, F.; Nesbitt, D. J. Quantum Deconstruction of the Infrared Spectrum of CH+ 5 . Science 2006, 311, 60–63.

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(21) Gnanasekar, S. P.; Arunan, E. Inter/Intramolecular Bonds in TH+ 5 (T = C/Si/Ge): H2 as Tetrel Bond Acceptor and the Uniqueness of Carbon Bonds. J. Phys. Chem. A 2019, 123, 1168–1176. (22) Okulik, N. B.; Peruchena, N. M.; Jubert, A. H. Three-Center-Two-Electron and FourCenter-Four-Electron Bonds. A Study by Electron Charge Density Over the Structure of Methonium Cations. J. Phys. Chem. A 2006, 110, 9974–9982. (23) Muller, H.; Kutzelnigg, W.; Noga, J.; Klopper, W. CH+ 5 : The Story Goes on. An Explicitly Correlated Coupled-Cluster Study. J. Chem. Phys. 1997, 106, 1863–1869. (24) Asvany, O.; Kumar, P.; Redlich, B.; Hegeman, I.; Schlemmer, S.; Marx, D. Understanding the Infrared Spectrum of Bare CH+ 5 . Science 2005, 309, 1219–1222. (25) Schmiedt, H.; Jensen, P.; Schlemmer, S. Unifying the Rotational and Permutation Symmetry of Nuclear Spin States: Schur-Weyl Duality in Molecular Physics. J. Chem. Phys. 2016, 145, 074301. (26) Schmiedt, H.; Jensen, P.; Schlemmer, S. Rotation-Vibration Motion of Extremely Flexible Molecules – The Molecular Superrotor. Chem. Phys. Lett. 2017, 672, 34 – 46. (27) Huang, X.; Johnson, L. M.; Bowman, J. M.; McCoy, A. B. Deuteration E↵ects on the Structure and Infra-Red Spectrum of CH+ 5 . J. Am. Chem. Soc. 2006, 128, 3478–3479. (28) Heck, A. J. R.; de Koning, L. J.; Nibbering, N. M. M. On the Structure of Protonated Methane. J. Am. Soc. Mass Spectom 1991, 2, 453–458. (29) Ivanov, S. D.; Asvany, O.; Witt, A.; Hugo, E.; Mathias, G.; Redlich, B.; Marx, D.; Schlemmer, S. Quantum-Induced Symmetry Breaking Explains Infrared Spectra of CH+ 5 Isotopologues. Nat. Chem. 2010, 2, 298–302. 23



(30) Anderson, J. B. A Random-Walk Simulation of the Schr¨odinger Equation: H+ 3 . J. Chem. Phys. 1975, 63, 1499–1503. 1 0 3 + (31) Anderson, J. B. Quantum Chemistry by Random Walk. H 2 P , H+ 3 D3h A1 , H2 ⌃u , 1 H4 1 ⌃ + g , Be S. J. Chem. Phys. 1976, 65, 4121–4127.

(32) Petit, A. S.; McCoy, A. B. Di↵usion Monte Carlo in Internal Coordinates. J. Phys. Chem. A 2013, 117, 7009–7018. (33) Suhm, M. A.; Watts, R. O. Quantum Monte Carlo studies of Vibrational States in Molecules and Clusters. Physics Reports 1991, 204, 293 – 329. (34) Langfelder, P.; Rothstein, S. M.; Vrbik, J. Di↵usion Quantum Monte Carlo Calculation of Nondi↵erential Properties for Atomic Ground States. J. Chem. Phys. 1997, 107, 8526–8535. (35) Burnham, K. P.; Anderson, D. R. Model Selection and Multimodel Inference; SpringerVerlag: New York, 2002. (36) Lin, M.; Lucas, H. C.; Shmueli, G. Research Commentary—Too Big to Fail: Large Samples and the p-Value Problem. Information Systems Research 2013, 24, 906–917. (37) Jones, E.; Oliphant, T.; Peterson, P. SciPy: Open Source Scientific Tools for Python. 2001; http://www.scipy.org/, [Online; accessed April 16, 2019].

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˚) Between Pairs of Hydrogen Atoms in CH+ Table 1: The Distances (in A 5 for the Equilibrium (EQ) Geometry and the Two Saddle Point (SP) Structures Shown in Figure 1.a Atomsb EQ C2v SP Cs SP 1, 2 2.03671 2.00649 1.54847 1, 3 1.44134 1.17849 2.01409 1, 4 1.79117 1.74851 1.85274 1, 5 1.79117 1.74851 1.75638 2, 3 0.95212 1.17849 0.94444 2, 4 1.71822 1.74851 1.84709 2, 5 1.71822 1.74851 2.01409 3, 4 1.94281 1.93834 1.84709 3, 5 1.94281 1.93834 1.54847 4, 5 1.87893 1.90212 1.85274 a Based on potential energy surface of Jin, Braams, and Bowman. 20 b Based on numbers on the hydrogen atoms in Figure 1

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FIGURES

Figure 1: The geometries of the minimum energy configuration, the C2v saddle point and the Cs saddle point based on the potential energy surface of Jin, Braams, and Bowman. 20

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2

CH5+ CH + 4

CH3D2+ CH D +

PHH (rHH)

PT(r)

2

a

5

CH4D+ CH D+

1.5

3 2

CH2D3+ CH D + 2 3

1

CHD4+ CHD + 4

CD5+ CD +

0.5

5

CH5+ CH5+

b

CH4D+ CH4D+

1.5

CH3D2+ CH3D2+ CH2D3+ CH D +

1

2 3

0.5 0

0 0.5 2

1.5

r (Å)

1Deut CH4D+ 2Deut CH3D2+ 3Deut CH2D3+ 4Deut CHD4+

1.5 1

0.5

2.5 2

c PDD (rDD)

PHD (rHD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.5

rHH (Å)

CH3D2+ CD3H2+ CH2D3+ CD4H+ CHD4+ CD5+ CD5+ CD2H3+

1.5 1

2.5

d

0.5

0.5

0

0 0.5

1.5

rHD (Å)

0.5

2.5

1.5

rDD (Å)

2.5

Figure 2: Projections of the ground state probability amplitude onto a) the distance between all of the pairs of hydrogen atoms, PT , b) the HH distances, c) HD distances, and d) DD distances in CHn D+ 5 n . The sticks in a) represent the distances between hydrogen atoms in the minimum energy structure shown in Figure 1, weighted by the number of hydrogenhydrogen distances that have that value. For these plots, we show the results for CH+ 5 with + + + + a purple line, CH4 D in blue, CH3 D2 in teal, CH2 D3 in green, CHD4 in yellow and CD+ 5 in red.

27



+ Figure 3: Projections of the ground state probability amplitudes for a) CH+ 5 , b) CH4 D , c) + + CH2 D3 , and d) CHD4 onto the indicated pairs of distances between H- or D-atoms. To aid in the visualization of the di↵erences, the probability amplitude is plotted on a log10 -scale.

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7.45

ln ℒ max /105 (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7.40

7.35

7.30

2

3

4

KDD

5

6

7

Figure 4: The values of the maximum log-likelihood ln Lmax obtained by fitting the PDD (rDD ) for CD+ 5 distributions to K Gaussian functions. The insets illustrate the fits, and are provided on a larger scale in Figures S9-S12 of the Supporting Information.

29



Figure 5: The values of the ck parameters, obtained from the fits to the distributions of distances between hydrogen atoms shown in Figure 2 which maximize the value of ln Lmax , are plotted for CH+ 5 and its deuterated isotopolgues. The color key used for this plot is defined in Figure 2. The values of the ck paramters shown in these plots are provided in Tables S7 to S10.

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Figure 6: The values of the ck parametersobtained when the PT (r) distributions for the six isotopologues of CH+ 5 shown in Figure 2(a) are fit to three Gaussian functions, K = 3 (left panel). In these plots, the black, red and blue circles represent the values of c1 , c2 and c3 , respectively. For comparison, the values of the ck parameters based on fits of the harmonic PT distributions for CH+ 5 (plotted in figures S1-S3) to three Gaussian functions are also provided in the right panel. The values of the ck parameters shown in these plots are provided in Tables S11 and S12.

31



TOC graphic 7.45

ln ℒ max /105 (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

7.40

7.35

7.30

2

3

4

KDD

32

5

6

7


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Statistical Analysis of the Effect of Deuteration on Quantum

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