Ion-neutral Collision Cross Section as a Function of the Static Dipole Polarizability and the Ionization energy of the Ion

Abstract

Ion mobility spectrometry is becoming more and more popular as a fast, efficient, and sensitive tool for the separation and identification of ionized molecules in gas phase. An ion traveling through a drift tube at atmospheric pressure under the influence of an electric field collides with the buffer gas molecules. The mobility of the ion depends inversely on the ion-neutral collision cross section. In the simplest hard-sphere approximation, the collision cross section is the area of the conventional geometric cross section. However, deviations are expected because of the physical interactions between the colliding species. More than a century ago, Langevin described a model for the interaction between a point-charge ion and a polarizable atom (molecule). Since then, the model has been modified many times to include better approximations of the interaction potential, usually preserving the point-charge nature of the ion. Although, more advanced approaches allow for considering polarizable ions with dissimilar sizes and shapes still explicit analytical dependencies on the properties of the ion remain elusive. In this work, an extended version of the Langevin model is proposed and solved using algebraic perturbation theory. A simple analytical expression of the collision cross section depending explicitly on both, the static dipole polarizability and the ionization energy of the ion is found. The equation is validated using ion mobility data. Surprisingly, even low-level calculations of the polarizability tensors produce results that are consistent with the experimental observations. This fact makes the equation very attractive for helping applications in different areas, such as the deconvolution of mobilograms of protomers, ion-molecule chemical kinetics, and others.

Graphical Abstract

Introduction

Ion-mobility spectrometry (IMS) is an analytical technique useful for separating and identifying ionized molecules in gas phase based on their mobility in a carrier buffer gas. Recently, IMS coupled to high-performance liquid chromatography and mass spectrometry has found many chemical and biological applications^{1, 2}.

IMS was first developed by Earl W. McDaniel of Georgia Institute of Technology in the 1950s and 1960s using drift cells with low applied electric fields to study ion mobilities and reactions in gas phase¹. The mobility, $\mathit{K}$, of an ion is defined by $\mathit{K}=\mathrm{v}/\mathrm{E}$ where $\mathrm{v}$ is the drift velocity and $\mathrm{E}$ is the electric field. The collision cross section of an ion, $\mathrm{\Omega }$, can be related to $\mathit{K}$ using kinetic theory^1–4 as shown in Eq. (1).

$$\mathrm{\Omega }=\frac{3\mathit{e}}{16\mathit{N}}\sqrt{\frac{2\mathit{\pi }}{\mathit{\mu }\mathit{k}\mathit{T}}}\frac{1}{\mathit{K}}$$ (1)

Where $\mathit{e}$ is the charge on the ion, $\mathit{N}$ the buffer gas number density at temperature $\mathit{T}$, and drift gas pressure $\mathit{P},\mathit{\mu }$ is the reduced mass of the pair buffer gas and ion, $\mathit{k}$ is the Boltzmann constant. $\mathrm{\Omega }$ is actually the momentum transfer collision integral^5–10, often called collision cross section (CCS), although a distinction has to be made between both⁵. Eq. (1) is known as Mason–Schamp Equation¹.

The quantity $\mathrm{\Omega }$ is generally very difficult to calculate theoretically, unless it is calculated for rigid spheres, in which case $\mathrm{\Omega }$ is equal to the projection cross section, $\mathrm{\Omega }=\mathit{\pi }{\mathit{b}}^2.\mathit{b}$ is called the impact parameter (see below). For a general geometrical shape, the projection cross section is only an approximation for the collision integral^{5–7, 10}.

The ion mobility is typically converted to a reduced mobility ${\mathit{K}}_0$ using a value of 760 Torr for ${\mathit{P}}_0$ and 273.15 K for ${\mathit{T}}_0$. ^2–4 Therefore, by measuring ${\mathit{K}}_0$ we can obtain a value for an experimental collision cross section.

As mentioned above, the ion-neutral interaction dynamics is extremely complicated and usually limited to capture models¹¹. The theoretical collision cross section for the interaction of an ion and a polarizable molecule having no permanent dipole moment (usually the case for buffer gases used in chemical applications) is approximated by the long-lived Langevin model¹² as extended by Gioumosis and Stevenson¹³. In this approach the ion-neutral collision dynamics is described using a two terms effective potential, ${\mathrm{V}}_{\mathit{e}\mathit{f}\mathit{f}}$, a repulsive centrifugal barrier $\left(\sim {\mathit{r}}^{-2}\right)$ and a long-range attractive potential $\left(\sim {\mathit{r}}^{-4}\right)$ as shown in Eq. (2):

$${\mathrm{V}}_{\mathit{e}\mathit{f}\mathit{f}}\left(\mathit{r}\right)=-\frac{1}{2}\frac{{\mathit{\alpha }}_0{\mathit{e}}^2}{{\mathit{r}}^4}+\frac{{\mathit{L}}^2}{2\mathrm{\mu }{\mathit{r}}^2}$$ (2)

${\mathrm{\alpha }}_0$ is the dipole polarizability of the buffer gas. Assuming ${\mathit{E}}_{\mathit{t}}$ is the translational energy in the center of mass frame, the ion-neutral relative velocity is $\mathrm{u}$ and the total relative energy is $\mathit{E}={\mathit{E}}_{\mathit{t}}+{\mathrm{V}}_{\mathit{e}\mathit{f}\mathit{f}}$ with ${\mathit{E}}_{\mathit{t}}=\mathrm{\mu }{\mathrm{u}}^2/2$, and conservation of angular momentum, $\mathit{L}=\mathrm{\mu }\mathrm{u}{\mathit{b}}_0$, where ${\mathit{b}}_0$ is defined as the shortest distance between the collision partners. When ${\mathit{b}}_0\ne 0$ the second term on the rhs of Eq. (2) leads to the above-mentioned centrifugal barrier. Thus the collision cross section, ${\mathrm{\Omega }=\mathit{\pi }{\mathit{b}}_0}^2$, can be derived from the conditions^{3, 14, 15}: $\mathit{d}{\mathrm{V}}_{\mathrm{e}\mathrm{f}\mathrm{f}}/\mathit{d}\mathit{r}=0$ and ${\mathrm{V}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathit{E}$ at $\mathit{r}={\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the position of the maximum of the centrifugal barrier. Finally, the analytical solution of the Langevin model is:

$${\mathrm{\Omega }=\mathit{\pi }{{\mathit{b}}_0}^2=\mathit{\pi }\mathit{e}\left(\frac{2{\mathit{\alpha }}_0}{\mathit{E}}\right)}^{1/2}$$ (3)

In this model, the ion is considered a point charge with no explicit dependence on its properties, other than the charge itself. Many modifications of the model have been performed during the years to include long-range contributions, such as ion-dipole interactions, quadrupolar interactions, etc^14–18, mostly on the side of the buffer gas. Mesleh et al., reported that the long-range interactions between the ion and the buffer gas molecules have a small, less than 10%, effect on the calculated ion mobility at room temperature. However, these effects are important if structural assignments are to be made from measured ion mobilities¹⁹. Long-range intermolecular interactions are usually described by a series expansion in powers of 1/r.

Eichelberger et. al. worked out a short, but comprehensive approach beyond the Langevin model to include polarizable ions²⁰. Following the procedure described above, the authors added a r⁻⁶-contribution to ${\mathrm{V}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ using indistinctly point polarizable ion model (PPI) or orientation dependent polarizable ion (OPPI) models to calculate the collision cross sections for several reactions of atomic oxygen with C_n⁻ and C_nH⁻ ions. It is also recommended to see references^{21, 22}. In fact, the algebraic equations involved in this or similar extended Langevin models can be solved analytically, however, the solutions are too cumbersome to be of any use in following ion mobility trends or comparing mobilities based on the properties of the ion. In this work, the dependence of collision cross section on the static dipole polarizability, the ionization energy of the ion and the relative orientation of the ion-neutral induced-dipoles is investigated using algebraic perturbation theory to obtain an analytical solution simple enough to be effectively used in such comparisons. The goal is to provide an equation with the right functional dependency, rather than a rigorous treatment of the problem.

Theory

As in previous work²⁰ we have developed an extension of the Langevin model by adding a long-range potential, ${\mathit{V}}_{\mathrm{\alpha }}$, to equation 2. A cartoonist representation of the interaction of a positive polarizable ion and a polarizable neutral molecule is shown in Figure 1.

Figure 1.

Left: Effective potential ${\mathrm{V}}_{\text{eff}}\left(r\right)$ (and its contributing terms) and the position $r_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of its maximum value, ${\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (centrifugal barrier). Right: At impact parameters below the maximum value, $b_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the ion-neutral interaction results in a short-range collision (or a chemical reaction in the case of a reactive collision). On the other hand, impact parameter values above $b_{\mathrm{m}\mathrm{a}\mathrm{x}}$ only produce small deflection of the collision pair.

Assuming the dominant r⁻⁶-contribution has the form of the asymptotic formula for the dispersion energy between two atoms at large distance $\mathit{r}$ known as London dispersion²³:

$${\mathit{V}}_{\mathrm{\alpha }}=-\frac{3{\mathit{I}}_0{\mathit{I}}_{\mathit{i}}}{2\left({\mathit{I}}_0+{\mathit{I}}_{\mathit{i}}\right)}\frac{{\mathrm{\alpha }}_0{\mathrm{\alpha }}_{\mathit{i}}}{{\mathit{r}}^6}$$ (4)

Where subscripts 0 and $\mathit{i}$ stand for the neutral buffer gas molecule and for the ion, respectively. $\mathrm{\alpha }$ is the static dipole polarizability and $\mathit{I}$ is the reduced ionization energy ${\mathit{I}}_0{\mathit{I}}_{\mathit{i}}/\left({\mathit{I}}_0+{\mathit{I}}_{\mathit{i}}\right)$. In the next sections, this reduced variable will be denoted simply by ${\mathit{I}}_{\mathit{i}}$. It is worth noting that ${\mathrm{V}}_{\mathrm{\alpha }}$ is attractive for any distance and that the inverse r⁻⁶-dependence on distance may contain several terms, e.g., the interaction between the charge of the ion and the quadrupolar moment of the molecule has a similar r⁻⁶-dependence on distance. This interaction is intense enough to be measured²⁴, therefore it has to be taken in account if the CCS data include different buffer gases with non-zero quadrupolar moments.

Analytical model

At first, any angular dependence of the ion-neutral London dispersion interaction is considered negligible as shown in Eq. (4) and the model is posed and solved in spherical approximation. Adding the contribution represented by Eq. (4) to Eq. (2), the effective potential is written as:

$${\mathrm{V}}_{\mathit{e}\mathrm{f}\mathrm{f}}\left(\mathit{r}\right)=-\frac{3{\mathit{I}}_{\mathit{i}}}{2}\frac{{\mathrm{\alpha }}_0{\mathrm{\alpha }}_{\mathit{i}}}{{\mathit{r}}^6}-\frac{1}{2}\frac{{\mathit{\alpha }}_0{\mathit{e}}^2}{{\mathit{r}}^4}+\mathit{E}\frac{{\mathit{b}}^2}{{\mathit{r}}^2}$$ (5)

Taking the derivative of Eq. (5), and setting it equal to zero we obtain:

$$9{{\mathit{I}}_{\mathit{i}}\mathrm{\alpha }}_0{\mathrm{\alpha }}_{\mathit{i}}+2{\mathrm{\alpha }}_0{\mathit{e}}^2{\mathit{r}}^2-2{\mathit{b}}^2\mathit{E}{\mathit{r}}^4=0$$ (6)

This equation can be analytically solved by hands or using a symbolic mathematical computation program; however, the solution is too complicated to be of any use. Instead, algebraic perturbation theory is applied in the usual way²⁵ to find an approximated value for the critical distance ${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ as follows: The highest order term is multiplied by a small parameter $\mathrm{ϵ}$ and the ion-neutral distance that maximizes the effective potential is written as,

$${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{\mathit{e}}{\mathit{b}}\sqrt{\frac{{\mathrm{\alpha }}_0}{\mathit{E}}}+\mathit{ϵ}{\mathit{r}}_1+{\mathit{ϵ}}^2{\mathit{r}}_2+{\mathit{ϵ}}^3{\mathit{r}}_3+\cdots$$ (7)

Where the first term of the rhs is the ${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ derived from the Langevin model and the rest terms are corrections for the London dispersion interaction conceived as a perturbation series of $\mathit{ϵ}$. Eq. (7) is substituted into Eq. (6) and the resultant equation expanded using the binomial theorem and then, the powers of $\mathit{ϵ}$ are matched up to find the corrections ${\mathit{r}}_1,{\mathit{r}}_2$, and so on.

At first order, the ${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ correction takes the simple form of the position of the Langevin barrier divided by two, so adding this to the Langevin value:

$${\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}=\frac{3}{2}\frac{\mathit{e}}{\mathit{b}}\sqrt{\frac{{\mathrm{\alpha }}_0}{\mathit{E}}}$$ (8)

Note that at first order ${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ shows no dependence on ${\mathrm{\alpha }}_{\mathit{i}}$. Putting this value of ${\mathit{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ into Eq. (5) and setting ${\mathrm{V}}_{\mathit{e}\mathit{f}\mathit{f}}=\mathit{E}$ we obtain an equation for $\mathit{b}$ (or ${\mathit{b}}^2$) that is also solved using algebraic perturbation theory (details of the procedure and the justification for these assumptions can be found with the supporting information). Finally solving for ${\mathit{b}}^2$ and putting into $\mathrm{\Omega }=\mathit{\pi }{\mathit{b}}^2$, we obtain a first order correction to the collision cross section as a function of the static dipole polarizability and the ion and neutral ionization energies:

$$\mathrm{\Omega }=\mathit{\pi }{{\mathit{b}}_0}^2+\frac{{\mathit{c}}_0\mathit{\pi }{\mathit{I}}_{\mathit{i}}{\mathrm{\alpha }}_0{\mathrm{\alpha }}_{\mathit{i}}}{{{\mathit{b}}_0}^4\mathit{E}}$$ (9)

As any perturbation theory approach is difficult to know $\mathit{a}$ priori if the series converges or diverges. It is worth noting that this equation is dimensionally correct, ${\mathit{b}}_0$-squared and static dipole polarizability has units of area, and volume, respectively. On the other hand, the ionization energy term has units of energy. Therefore, the second term of equation 9 has unit of area as expected.

Because of its simplicity, equations 9 may be very useful for practical applications. The adjustable parameter ${\mathrm{c}}_0$ in Eq. (9) can be found, either by comparison with the exact solution of Eq. (6) or using the best-fit to experimental collision cross section data. The theoretical value emerging from the series truncation is 8/21.

Methods

Experimental ion mobilities were taken from the literature²⁶. Although there are other experimental values available, this set was chosen for three main reasons, 1) it contains a good number of dissimilar molecules (94), 2) all ion mobilities were determined under the same conditions, e.g., the buffer gas in all measurements is air 3) these are protonated amines or nitrogen-containing heterocycles that usually have well-defined protonation sites. This latter is important to avoid uncertainty in the polarizability calculations. The fact that these measurements were performed using air instead of molecular nitrogen might introduce some uncertainty in the results.

The static dipole polarizabilities were calculated using the hybrid density functional method B3LYP^{27, 28} in conjunction with the Dunning’s correlation consistent basis set cc-pVTZ²⁹ as implemented in Gaussian 09³⁰. Accurate calculations of the polarizability tensor probably require larger basis sets; however, it has been shown before that density functional theory produces reasonable polarizabilities with intermediate-sized basis sets when using exchange-correlation functionals with the correct asymptotic behavior³¹. Vibrational frequency analysis at the same level of theory was used to identify the optimized structures as minima.

Results and Discussion

As stated before, the ion-neutral collision dynamics is very complicated, so the goal was to obtain equations which describe the correct functional dependence of CCS on some ion properties in a way that the results can be used intuitively or performing only simple calculations.

Table 1 shows the ion mobilities, the reduced ionization energies, the diagonal components of the polarizability tensors, and the static dipole polarizabilities of the ions. Ionization energies were estimated using the Koopman’s theorem³². Static dipole polarizabilities were calculated as a third of the trace of the polarizability tensor. Other quantities derived from Eq. (9) are not shown but can be easily calculated from the data in Table 1.

Table 1.

Experimental and DFT-calculated^* properties of protonated nitrogen-containing compounds. Ion mobilities (cm²/(V s), molecular weights (amu), ionization energies (eV), and static dipole polarizabilities (10⁻²⁴cm³). Data for the nitrogen molecule is included in the last row of the table.

Name	MW	$K_0$	$\mathrm{I}$	$\mathrm{\alpha }$
1–2-phenylenediamine	108	1.96	11.43	12.06
1–3-phenylenediamine	108	1.87	9.75	12.78
1–4-phenylenediamine	108	1.82	9.69	12.82
1c2c4-triazole	69	2.21	14.15	5.55
1-naphthylamine	143	1.74	10.20	18.22
2–2-dipyridyl	156	1.71	11.07	18.76
2–3-dimethylpyrazine	115	1.89	12.32	11.96
2–5-dimethylpyrazine	108	1.90	12.34	12.14
2-chloroaniline	127	1.79	11.42	12.06
2-ethylaniline	121	1.74	11.13	13.84
2-isopropylaniline	135	1.66	11.08	15.62
2-phenylpyridine	155	1.64	10.81	19.50
2-picoline	93	2.03	12.61	10.08
2-toluidine	107	1.83	11.21	12.01
3-chloroaniline	127	1.86	11.06	12.24
3-cyanopyridine	104	1.91	13.04	10.63
3-hydroxypyridine	95	2.06	12.12	9.01
3-picoline	93	2.00	12.59	10.06
4–4-dipyridyl	156	1.60	10.42	18.46
4-amino-N-N-diethylaniline	164	1.58	8.90	20.02
4-hydroxypyridine	95	2.06	12.62	8.91
4-phenylpyridine	155	1.64	10.59	20.01
4-picoline	93	2.07	12.92	10.09
4-tert-butylpyridine	135	1.70	12.49	15.48
4-toluidine	107	1.95	11.12	12.20
Acridine	179	1.59	10.45	23.99
Aniline	93	1.93	11.60	10.10
Benzyldimethylamine	135	1.71	10.70	15.62
Bromoquinoline	208	1.63	11.33	19.07
Cinnoline	130	1.82	11.62	14.78
Cyclohexylamine	99	1.83	12.53	10.92
Diaminoethane	60	2.25	10.95	5.89
Dibenzylamine	197	1.41	10.39	23.48
Dicyclohexylamine	181	1.44	12.06	20.82
Didodecylamine	353	0.89	9.7177	45.71
Diethanolamine	105	1.91	11.88	9.36
Diethylamine	73	2.16	14.61	8.12
Diethylaniline	149	1.67	11.12	17.15
Diisobutylamine	129	1.66	13.12	15.25
Diisopropylamine	101	1.92	13.99	11.53
Dimethylamine	45	2.46	16.48	4.53
dimethylamino-1-propane	101	1.94	13.48	9.73
N-N-dimethylformamide	73	2.21	13.99	6.36
di-n-butylamine	129	1.64	12.57	15.54
di-n-hexylamine	185	1.34	11.41	23.09
di-n-propylamine	101	1.87	13.43	11.80
Diphenylamine	169	1.56	11.02	19.54
Ethanolamine	61	2.23	12.17	5.25
Ethylamine	45	2.36	15.05	4.73
Formamide	45	2.45	15.80	3.11
Hexamethylenediamine	119	1.77	8.57	13.20
Hexamine	140	1.68	10.90	12.31
Imidazole	68	2.29	12.61	6.09
Indoline	121	1.92	11.18	12.98
Isobutylamine	73	2.02	13.55	8.13
Isopropylamine	59	2.20	14.72	6.38
Isoquinoline	129	1.85	11.09	15.23
Methylamine	31	2.65	17.18	2.90
Morpholine	87	2.00	11.04	8.09
n-butylamine	73	1.98	12.85	8.26
n-decylamine	157	1.35	10.24	19.41
n-dodecylamine	185	1.26	9.81	23.15
N-ethylaniline	121	1.79	11.28	13.73
n-heptylamine	116	1.61	11.18	13.81
n-hexadecylamine	241	1.14	9.23	30.65
n-hexylamine	101	1.72	11.59	11.95
N-methylaniline	107	1.95	11.41	11.84
N-methylimidazole	82	2.19	12.34	7.72
N-N-dimethylacetamide	87	2.09	13.33	8.37
N-N-dimethylaniline	121	1.81	11.25	13.52
N-N-dimethylcyclohexylamine	129	1.73	12.21	14.20
N-N-Dimethylpropanediamine	102	1.94	11.44	11.82
n-nitroaniline	138	1.78	11.61	13.00
n-octylamine	129	1.51	10.85	15.68
n-pentylamine	87	1.85	12.13	10.10
n-propylamine	59	2.14	13.80	6.44
n-tetradecylamine	213	1.20	9.48	26.90
o-nitroaniline	138	1.78	12.39	12.76
Phthalazine	130	1.82	11.76	14.38
Pyrazine	80	2.22	12.98	7.49
Pyridazine	80	2.07	13.97	7.38
Pyridine	79	2.21	13.19	8.03
Quinoline	129	1.82	11.33	15.22
Quinoxaline	130	1.80	11.69	14.81
Salicylamide	137	1.81	10.78	13.87
sec-butylamine	73	2.06	13.67	8.13
tert-butylamine	73	2.03	14.38	8.07
tert-pentylamine	87	1.95	13.69	9.75
Triazine	81	1.95	13.41	6.60
Triethylamine	101	1.95	14.01	11.35
Triisooctylamine	353	0.93	10.69	44.65
Trimethylamine	59	2.36	16.10	6.17
tri-n-butylamine	185	1.38	12.39	22.44
tri-n-hexylamine	269	1.11	11.31	33.75
Molecular nitrogen	31	N/A	7.79	1.43

^*Molecules were optimized using DFT at B3LYP level with the cc-pVTZ basis set requiring the analytic second derivatives be calculated at every step in the geometry optimization. Polarizabilities were calculated at the same level of theory.

Mason–Schamp equation, i.e., Eq. (1), can be used in conjunction with Eq. (9) for testing the validity of the extended Langevin model. Figure 2 shows a plot of ${\left({\mathrm{\mu }}^{1/2}{\mathit{K}}_0\right)}^{-1}$ as functions of $\mathrm{\alpha }$. The equation is in good agreement with the experimental data with coefficient of determination $\left({\mathrm{R}}^2\right)$ of 0.91.

Figure 2.

Plot of ${\left({\mathrm{\mu }}^{1/2}{K}_0\right)}^{-1}$ as functions of $\mathrm{\alpha }.K_0$ (cm²/(V s)), $\mathrm{\alpha }$ (10⁻²⁴ cm³).

The ionization energy of the nitrogen molecule, a typical buffer gas, is 15.58 eV, which is comparable to that of protonated molecules (see Table 1). Therefore, the dependence of the collision cross section on ionization energy is very weak. Using the data in Table I shows that in the Koopman’s approximation, ${\mathit{I}}_0{\mathit{I}}_{\mathit{i}}/\left({\mathit{I}}_0+{\mathit{I}}_{\mathit{i}}\right)$ is approximately constant and equal to 6.77±0.05 eV. However, ionization energy and polarizability are not completely independent. Non-linear relationships between ionization energy and polarizability have been reported before^{33, 34}. Therefore, some contribution to the scatter of individual data points can be expected. Overall, the effect of ionization energy on the theoretical collision cross section is minor.

Perhaps, the most remarkable aspect of the results presented in this paper is the fact that the plots derived from Eq. (9) hold when using more complex ions and semiempirical polarizability tensors. Semiempirical methods are computationally cheap, however, they are not very accurate regarding polarizability calculations, due to minimal basis set incompleteness. The polarizability tensors of a group of typical metabolites with known collision cross sections³⁵, Table 2, were calculated using the semiempirical method PM6³⁶. Yet another set of mycotoxins³⁷ with multiple protonation (or deprotonation) sites were calculated at the same level of theory and the results included with the supplementary information (see Table 1S and Table 2S). All molecules were optimized requiring the analytic second derivatives be calculated at every step in the geometry optimization before the polarizability calculation. The relevant data is shown in Table 2.

Table 2.

Experimental collision cross sections and PM6-calculated properties of protonated metabolites, [M+H]⁺ ion, in positive mode. Static dipole polarizabilities (10⁻²⁴ cm³), highest occupied molecular orbital energy (eV), and collision cross sections (cm²).

Name	$\mathrm{\alpha }$ (10−24 cm3)	EHOMO (eV)	$\mathrm{\alpha }$ * $\mathrm{I}$	CCS (cm2)
1-Methylhistidine	12.70	−13.27	168.50	129
2-Phosphoglyceric Acid	9.34	−15.26	142.52	132
3-Phosphoglyceric Acid	9.23	−15.17	140.02	134
5-Oxoproline	9.14	−14.56	133.10	121
Acadesine	18.29	−12.75	233.09	152
Acetylcarnitine	12.78	−14.32	182.95	142
Adenosine	20.15	−13.23	266.70	151
ADMA	14.58	−11.35	165.52	141
Arginine	11.59	−12.72	147.35	133
Aspartic Acid	7.76	−15.82	122.72	122
Carnitine	9.90	−14.85	146.93	130
Choline	6.59	−14.83	97.79	117
Citric Acid	11.33	−15.02	170.21	138
Citrulline	11.07	−13.38	148.05	132
Cytidine	17.07	−13.29	226.83	148
Dimethylglycine	6.32	−15.81	99.86	116
Epinephrine	13.89	−11.97	166.32	133
Fructose	10.49	−14.46	151.69	135
Galactose	10.25	−14.16	145.16	136
Glucosamine	10.39	−13.80	143.27	135
Glucose	10.51	−14.64	153.84	138
Glucuronic Acid	11.10	−14.32	158.95	140
Glutamic Acid	8.58	−15.57	133.56	123
Glutamine	9.31	−13.49	125.69	124
Glycerol monophosphate	8.52	−14.50	123.55	132
Guanosine	21.54	−12.76	274.92	160
Histidine	10.89	−13.25	144.26	127
Homocysteine	10.80	−12.28	132.58	124
Hydroxyproline	7.77	−15.15	117.76	122
Inosine	20.03	−13.63	272.99	153
Isoleucine	8.24	−15.41	126.96	129
Leucine	8.46	−15.39	130.20	128
Lysine	9.41	−12.38	116.46	127
Mannitol	10.14	−13.53	137.26	137
Mannose	10.69	−14.70	157.18	138
Methionine	12.17	−12.03	146.47	127
Mevalonic Acid	8.97	−14.30	128.24	128
Ornithine	8.34	−13.18	109.92	121
PABA	10.57	−14.36	151.84	131
Pantothenic Acid	13.63	−13.53	184.34	145
Phenylalanine	12.41	−12.65	157.07	135
Phosphoenolpyruvate	9.62	−14.92	143.45	131
Phosphorylcholine	10.17	−14.90	151.52	135
Proline	7.18	−15.71	112.75	118
Pyridoxamine	12.73	−12.93	164.60	130
SDMA	13.63	−12.77	174.07	142
Threonine	6.83	−15.18	103.75	122
Tryptophan	16.40	−11.42	187.34	145
Tyrosine	13.97	−12.37	172.87	139
Uridine	17.54	−13.83	242.57	152
Xanthosine	21.15	−13.61	287.84	158

^*Molecules were optimized using the semiempirical method PM6 and calculating the second derivative at each point. Polarizabilities were calculated at the same level of theory.

As suggested by the previous analysis, a linear relationship is obtained when plotting the collision cross section as a function of the static dipole polarizability. The coefficient of determination $\left({\mathrm{R}}^2\right)$ is 0.86 (more information can be found in the supplementary information (Excel spreadsheet: CCS_vs_polarizability_metabolites_positive-ions_for_Table_2.xlsx, Fig. S1). Although some effect of the ionization potential is observed it is almost negligible. The behavior of negative ions, [M-H]-, of the same set of metabolites is very similar (supplementary information, Excel spreadsheet: CCS_vs_polarizability_metabolites_negative-ions.xlsx, Fig. S2). For the sake of simplicity, we have ignored some improvement in the correlation obtained by including the anisotropy of the polarizability tensor in Eq. (5). The semiempirical calculations involve CPU-times in the range 0–2 seconds each.

Despite Eq. (9) provides a way to predict reasonable CCS values by minimizing c₀ using experimental data, this is not the main purpose of this approach. There are software available that calculate CCS values based on less constrained approximations, e.g., IMoS, Collidoscope, Mobcal, etc. The advantage of the present model is that it is a model with analytical solution that is susceptible of improvement and that can help to understand the contributions of the multipolar expansion terms to ion mobility. In this work, it has been established that the static dipole polarizability strongly correlates with the observed CCS values. In practical terms, polarizability can be intuitively used to analyze trends without any calculations. It can also be very useful for the interpretation, deconvolution, and identification of isomers from low-resolution ion mobility spectra involving multiple isobaric ions or protomers.

Equation 9 of the manuscript suggests that the absolute differences of the experimental CCS values of protomers must be proportional to the corresponding absolute differences of polarizabilities, $\left|{\mathrm{\Omega }}_{\mathrm{i}}-{\mathrm{\Omega }}_{\mathrm{j}}\right|\mathrm{\alpha }\left|{\mathrm{\alpha }}_{\mathrm{i}}-{\mathrm{\alpha }}_{\mathrm{j}}\right|$. In other words, the larger the changes in protomer polarizabilities, the larger the changes in collision cross-sections. To validate this assumption, the absolute differences of CCS values of protomers of four molecules, with one with two or more different protonation sites (Aniline, Melphalan, p-Benzocine, and m-Benzocaine), and reported in a recent study³⁸ were plotted against the absolute differences of static dipole polarizabilities calculated using DFT calculations as described in the section Methods. As expected, the experimental values are highly correlated with the calculated polarizability differences (coefficient of determination, R² = 0.99, see supplementary information, Excel spreadsheet: Protomers_Delta_CCS_versus_Delta_alpha.xlsx, Fig. S3). This by itself shows that the changes in protomer CCs values are determined by the corresponding polarizabilities. Although protomer studies are scarce, protomers have a significant influence on tandem mass spectra³⁹ and are worth studying.

It is worth noting that although successful in practice, state-of-the-art theoretical descriptions of the collision processes involve potentials that contain empirical quantities that cannot be measured or derived from physical observables, such as 6–12 Lennard-Jones potentials, molecular shapes, and others. These approaches are generally used as black boxes and the uncertainty of the CCS values difficult to estimate. On the other hand, the resolution of the current ion mobility instrumentation is still insufficient regarding certain problems. Table 2 shows several structurally different protonated molecules having the same CCS value, e.g., 1-Methylhistidine (129) and Isoleucine (129).

In summary, the simple structure of Eq. (9) offers new insights into the ion-neutral interactions, and it is easy to extend to other areas such as ion-molecule chemical reactions. Also, the average static dipole polarizability is related to other properties of the ion, such as the HOMO-LUMO gap^{40, 41} or the dimensions of the ion⁴², so the ion mobility analysis can be extended to include these properties if they are available. The approach provides a very intuitive framework that can help chemists with ion mobility and ion-molecule reaction data.

It is worth to mention that a recent review reevaluates the role of polarizability in ion mobility spectrometry⁴³. Unfortunately, we were not aware of this publication at the time our manuscript was submitted.

Conclusion

Algebraic perturbation theory has been used to derive an equation for the collision cross section (CCS) of the interaction between a polarizable ion and a polarizable neutral molecule as a function of both, the static dipole polarizability, and the ionization energy of the ion using an extended version of the Langevin model. The obtained equation shows a linear dependence between ion mobility and the ion static dipole polarizability. On the other hand, the effect of the ionization energy for the compounds used in this work is almost constant. However, the ionization energy can influence differently individual points of the plot due to its inverse relationship to static dipole polarizability. In summary this extended version of the Langevin model shows the correct functional dependence on the ion static dipole polarizability and a good agreement with the experimental ion mobilities. As stated in the section Results and Discussion, the present model can be useful for three reasons: a) it has an analytical solution, b) it is susceptible to improvement, and c) it preserves the physical meaning of the quantities involved.

ABBREVIATIONS

IMS

Ion mobility

CCS

Collision cross section

Polarizability

Polarizability tensor