Theoretical Calculation of Polarizability Isotope Effects

Journal of Molecular Modelling manuscript No. (will be inserted by the editor)

Theoretical Calculation of Polarizability Isotope Effects F´ elix Moncada · Roberto Flores-Moreno · Andr´ es Reyes

Received: date / Accepted: date

Abstract We propose a scheme to estimate hydrogen isotope effects on molecular polarizabilities. This approach combines the any particle molecular orbital method, in which both electrons and H/D nuclei are described as quantum waves, with the auxiliary density perturbation theory, to calculate analytically the polarizability tensor. We assess the performance of method by calculating the polarizability isotope effect for 20 molecules. A good correlation between theoretical and experimental data is found. Further analysis of the results reveals that the change in the polarizability of a X-H bond upon deuteration decreases as the electronegativity of X increases. Our investigation also reveals that the molecular polarizability isotope effect presents an additive character. Therefore, it can be computed by counting the number of deuterated bonds in the molecule. Keywords Isotope effect · Polarizability · Non covalent interactions · Nuclear orbital · Density Functional Theory.

1 Introduction In recent years there has been an increasing interest in the study of H/D isotope effects on various phenomena where non-covalent interactions [1] are dominant such as: chromatography [2–6], sorption [7–9], host-guest interactions [10–12], proteinligand binding [13, 14], among others. Non-covalent interactions are governed by electrostatic interactions and van der Waals forces. The magnitude of the latter can be estimated from molecular F. Moncada Chemistry Program, Universidad de la Amazonia, Calle 17 Diagonal 17 - Carrera 3F, Florencia, Colombia E-mail: [email protected] R. Flores-Moreno Department of Chemistry, Universidad de Guadalajara, Blvd. Marcelino Garc´ıa Barrag´ an 1421, Guadalajara Jal., C.P. 44430, Mexico E-mail: [email protected] A. Reyes Department of Chemistry, Universidad Nacional de Colombia, Av. Cra. 30 #45-03, Bogot´ a, Colombia E-mail: [email protected]

2

F´ elix Moncada et al.

polarizabilities α employing the Casimir-Polder equation [15]. As a consequence, H/D isotope effects on the strength of non-covalent interactions will highly depend on the polarizability isotope effects (PIEs). The origin of PIEs was associated to the bond length contraction that occurs upon deuteration [16]. In this regard, it has been established that the shorter length of the X-D bond results in lower polarizabilities [16]. To date, experimental information on PIEs is only limited to about 30 compounds [17–22]. To the best of our knowledge, there is a limited number theoretical studies on molecular PIEs. However, the studies either include experimental information [23] or are limited to the H2 molecule [19]. The lack of theoretical studies on PIEs must be due to the need of evaluating anharmonicity of the potential experienced by the H/D atom, which is computationally expensive due to the need of third and higher order energy derivatives. Nuclear molecular orbital (NMO) methods have been proposed recently as a way to recover nuclear quantum effects on electronic structure and molecular properties [24–29]. These effects are obtained directly from one single electronic/nuclear calculation and not as further corrections. NMO methodologies have been used to analyze H/D isotope effects in a wide variety of molecular phenomena [26–41]. We developed the any-particle molecular orbital (APMO) method, which extends the NMO formalism to any type of quantum species [27]. Later, we incorporated the auxiliary density functional theory (ADFT) formalism within the APMO approach [42]. Our aim in this paper is to propose a methodology that combines the APMO approach with the auxiliary density perturbation theory (ADPT) [43, 44] to calculate theoretically molecular PIEs. The ADPT allows for the analytic calculation of the polarizability and other response properties. We will refer to this new methodology as APMO/ADPT. This paper is organized as follows: In the theoretical aspects section we summarize the expressions of APMO and ADPT methods required to perform PIE calculations. In the computational details section we describe the methodology used in APMO/ADPT calculations. In the results and discussion section we present a benchmark of the APMO/ADPT PIEs and a study of the PIE trends for different molecules and bonds. In the conclusions section we provide final remarks.

2 Theoretical aspects Here, we summarize the equations of the APMO method, including the ADFT formalism [42], along with those of the electronic ADPT method [43].

2.1 Any Particle Molecular Orbital plus Auxiliary Density Functional Theory The APMO method analyzes molecular systems comprising any number of quantum species [27]. In this work, we employ a subset of the APMO equations, in which electrons, H and D nuclei are treated as quantum waves [42] while atomic nuclei with Z ≥ 2 are considered as point-charges under the Born-Oppenheimer approximation. Electrons are described using the ADFT [45], whereas the wavefunctions of H and D nuclei are represented with localized Hartree products [46].

Theoretical Calculation of Polarizability Isotope Effects

3

The Hamiltonian for a system comprising Ne electrons, in closed-shell configuration, Nn quantum nuclei and NC point charge nuclei is Htot = −

Ne X 1

2

i

∇2i −

Nn X α

Ne NC

Nn NC

X X qJ X X qα qJ 1 ∇2α − + 2Mα riJ rαJ α i

J

I

Nn Ne X NC Ne Nn X X X qα qβ X qI qJ 1 qα + − + . + rij rαβ r rIJ iα α j>i

i

β>α

(1)

J>I

The corresponding Kohn-Sham energy functional, E, of this system is KS [ρe ] + E[ρe , {ρα n }] =Te

Nn X

TnKS [ρα ] +

Z ρe (r)v(r)dr −

α

ZZ +

−

Nn X α

ZZ

Z qα

ρα n (r)v(r)dr

α Nn X

ρe (r)ρe (r’) drdr’ + | r − r’ | qα

Nn X

ZZ qα qβ

β>α

β ρα n (r)ρn (r’)

| r − r’ |

drdr’

ρα n (r)ρe (r’) drdr’ + Exc [ρe , {ρα n }], | r − r’ |

(2)

KS and TnKS here, ρe is the electronic density, {ρα n } is the set of nuclear densities, Te are the electronic and nuclear Kohn-Sham kinetic energy functionals, v(r) is the external potential and Exc is the global exchange-correlation energy functional. The Exc functional includes electronic exchange-correlation, nuclear-electron correlation, and nuclear-nuclear exchange-correlation contributions. In this work, we e [ρe ] only consider the electronic exchange-correlation contribution, Exc e Exc [ρe , {ρα n }] ≈ Exc [ρe ].

(3)

Electronic and nuclear densities are calculated as ρe (r) =

Ne X

| ψie (r) |2 ,

i

ρα n (r)

= | ψ1α (r) |2 ,

(4)

for α = 1, 2, . . . Nn . Considering that quantum nuclei are considered as distinguishable particles, each nuclear orbital is occupied by one nucleus. Kohn-Sham orbitals, {ψ}, are expanded in terms of Gaussian-type functions (GTFs), {θ} ψie (r) =

Be X

ceµi θµe (r)

µ

ψiα (r) =

Bα X

α cα µi θµ (r),

µ

where Be and Bα are the electronic and nuclear α basis set sizes, respectively.

(5)

4


The canonical extended Kohn-Sham equations for electrons and nuclei are ! Z Z Nn 0 X ∇2i ρe (r0 ) ρα 0 0 e n (r ) − + v(r) + dr − qα dr + vxc [ρe ] ψie = ei ψie 0 | 2 | r − r0 | | r − r α ! Z Z Nn 2 0 X ∇α ρe (r ) ρβn (r0 ) 0 0 − − qα v(r) − qα dr + qα qβ dr ψ1α = α ψ1α , 2Mα | r − r0 | | r − r0 | β6=α

(6) e for i = 1, 2, . . . , Ne and α = 1, 2, . . . Nn . Here, vxc is the electronic exchangecorrelation potential. The evaluation of the molecular energy is simplified by invoking the variational approximation of the Coulomb potential for electrons [45, 47–52]. First, we ˜ introduce the electronic auxiliary density expanded in terms of GTFs, {θ}

ρ˜e (r) =

Ae X

xek θ˜ke (r),

(7)

k

where Ae is the auxiliary basis set size and {xe } is a set of electronic fitting coefficients obtained by minimizing the self-interaction error [47, 49], ZZ 1 [ρe (r) − ρ˜e (r)][ρe (r0 ) − ρ˜e (r0 )] ξ2e = drdr0 . (8) 2 | r − r0 | Defining the elements of the Coulomb matrix, G, and the Coulomb vector, J, [45, 52] as Gekl ≡ hθ˜ke | θ˜le i Jke ≡

Be X

(9)

e hθµe θνe | θ˜ke i, Pµν

(10)

µν

a linear equation system can be formulated for the determination of the fitting coefficients for electrons (xe ) as xe = (Ge )−1 Je .

(11)

A Kohn-Sham energy expression can be written in terms of GTFs by employing the auxiliary density in the evaluation of the exchange-correlation functional [52] E=

Be X

e e Pµν Hµν +

α

µν

−

Nn X Be X Bα X α

+

Nn X Bα X µν

α α Pµν Hµν +

Ae X

Ae

Jk xek −

k

1X e e xk xl Gkl 2 kl

e α qα Pµν Pστ hθµe θνe | θσα θτα i

µν στ

Bβ Nn X Bα X X

α β e qα qβ Pµν Pστ hθµα θνα | θσβ θτβ i + Exc [ρ˜e ].

(12)

β>α µν στ e α Here, we used Mulliken’s notation for Coulomb integrals [53]. Pµν and Pµν are the e α electron and nucleus α density matrices elements; Hµν and Hµν are the electronic


5

and nuclear α core matrices elements. Expressions for Pµν and Hµν are defined as in regular electronic structure methods [53]. Kohn-Sham matrices elements are obtained by expressing the ADFT energy in terms of electronic and nuclear density matrix elements e e Kµν =Hµν +

Ae X

(xek + zke )hθµe θνe | θ˜ke i − 2

α

k α α Kµν =Hµν +

Nn X Bα X

Bβ Nn X X

(13)

στ

β qα qβ Pστ hθµα θνα | θσβ θτβ i −

β>α στ

α qα Pστ hθµe θνe | θσα θτα i

Be X

e qα Pστ hθµα θνα | θσe θτe i,

(14)

στ

where the exchange-correlation coefficients are defined as zke ≡

Ae X

˜e e (Ge )−1 kl hθl | vxc [ρ˜e ]i.

(15)

l

2.2 Electronic Auxiliary Density Perturbation Theory In the framework of the self-consistent perturbation theory [54], the molecular polarizability tensor is calculated in terms of second derivatives of the energy with respect to an external field. These derivatives are calculated employing the ADFT. These calculations require the evaluation of the perturbed density matrix. We refer to this methodology as ADPT [43]. Given the perturbation independent basis and auxiliary functions, the first order closed-shell perturbed density matrix elements are

oc un

e(λ) Pµν =2

e X X K e(λ) e e ∂Pµν e e ia =2 e − e (cµi cνa + cµa cνi ). ∂λ a i a

(16)

i

Here, λ is a perturbation parameter, ei and ea are orbital energies of the ith e(λ) occupied and ath unoccupied orbital, and Kia is an element of the perturbed Kohn-Sham matrix in the molecular orbital representation: e(λ)

Kia

=

Be X

e(λ) , ceµi ceνa Kµν

(17)

µν e(λ)

here, Kµν is a perturbed Kohn-Sham matrix element in the atomic orbital representation. We assumed that the deformation of the nuclear densities caused by the perturbation is negligible, P α(λ) = 0. A similar approximation was employed by Tachikawa and coworkers in the calculation of isotope effects on NMR shielding e(λ) tensors [41, 40]. The perturbed density matrix elements, Pµν , are obtained by

6


first taking the derivative of Eq. 13 with respect to the perturbation parameter and then replacing it in Eq. 16 e(λ) Pµν =2

oc X un e(λ) X Hia ceµi ceνa + ceµa ceνi e e − a i a i

+2

Ae oc X un X X hψie ψae | θ˜ke i e(λ) e e xk cµi cνa + ceµa ceνi e e i − a a i

k

Ae oc X un X X hψie ψae | θ˜ke i ee e(λ) e e cµi cνa + ceµa ceνi , Fkl xl +2 e − e a i a i

(18)

kl

e(λ)

here, Hia is the perturbed core Hamiltonian matrix in the molecular orbital representation, which gathers operator derivatives, hψie ψae | θ˜ke i is a three-center e(λ) repulsion integral in the molecular orbital representation, xk is a perturbed ee fitting coefficient and Fkl is an element of the kernel matrix, evaluated as follows: ee Fkl ≡

X

e ˜e ρ] | θ˜le i, (Ge )−1 km hθm | fxc [˜

(19)

m ee is the electron-electron exchange-correlation kernel. where fxc e(λ) The set of xk is obtained by deriving the fitting equation (Eq. 11) with respect to the perturbation parameter Be X

e(λ) e e ˜e Pµν hθµ θν | θm i =

µν

Ae X

e(λ)

Gekm xk

.

(20)

k

Inserting Eq. 18 into Eq. 20 we obtain X Ae oc X un ˜e oc X un X e(λ) e X hθm | ψie ψae iHia hθ˜m | ψie ψae ihψie ψae | θ˜ke i e(λ) 4 + 4 xk e i − ea ei − ea a a i

i

+4

k

Ae oc X un X X i

a

kl

X Ae e | ψie ψae ihψie ψae | θ˜ke i ee e(λ) hθ˜m e(λ) x F = Gekm xk . kl l ei − ea k

(21) We define the perturbation independent Coulomb matrix, A, the exchangecorrelation matrix, B, and the perturbation vector, bλ . Their elements are constructed as oc X un ˜e X hθl | ψie ψae ihψie ψae | θ˜ke i ei − ea a i X e ee ≡ Akm Fml

Aekl ≡ e Bkl

e(λ)

bk

≡

m oc X un X i

a

(22) (23)

e(λ)

e hθ˜m | ψie ψae iHia ei − ea

.

(24)

Theoretical Calculation of Polarizability Isotope Effects e(λ)

The set of xk

7

is determined by solving the system of linear equations: xe(λ) = 4(Ge − 4Ae − 4Be )−1 be(λ) .

(25)

e(λ)

Perturbed density matrix elements, Pµν , are calculated with the perturbed fitting coefficients using Eq. 18. In polarizability calculations the components of the applied static electric field vector, E = (Ex , Ey , Ez ) are used as the perturbation parameters, λ. We build three perturbation vectors, be(Ex ) , be(Ey ) , be(Ez ) , employing dipole moment integrals as perturbed core Hamiltonian matrices, e(Ex )

= hψie | x ˆψae i

e(Ey )

= hψie | yˆψae i

Hia

Hia

e(Ez )

Hia

= hψie | zˆψae i.

(26)

The unperturbed electronic and nuclear molecular orbitals are coupled in the APMO approach. As a consequence, electronic properties depend on nuclear masses and isotope effects on molecular polarizability can be studied with the ADPT methodology.

3 Computational details The electronic ADPT method was implemented in the LOWDIN software [55]. APMO/ADPT calculations performed with this software considered electrons and H/D nuclei as quantum particles and heavier nuclei as point charges. Calculations were performed under the local density approximation (LDA), with the electronic Slater-Dirac exchange [56, 57] and Vosko-Wilk-Nusair correlation [58] functionals. Previous DFT studies employing those functionals have reported errors of 5% with respect to experimental values in the polarizabilities of small to medium size molecules [59, 60]. Calculations were performed with the Def2-TZVPD electronic basis set because polarizabilities produced with this basis set are very close to the basis set limit [61]. Electronic auxiliary basis set was built with primitive Cartesian GTFs using the GEN-A2* [62] algorithm. Numerical integration of exchange-correlation kernel, potential and energy were performed employing a (75,302) fixed grid. Convergence in SCF calculations was achieved when total energy changes were lower than 10−9 a.u.. We analyzed the effect of the nuclear basis set size on α and PIE. To that aim we built different nuclear basis sets using the even-tempered scheme, i

ζi = γ × 10 2

i=−

N −1 N −1 , . . . 0, . . . , . 2 2

Here, ζi is a reference exponent and i is an integer that depends on the chosen number of GTFs, N . γ values were obtained by optimizing the exponents of a 1s1p1d nuclear basis for H2 and D2 molecules with the APMO/HF method [27]. These values are reported in Table 1. Nuclear basis set functions were placed at the positions occupied by the H atoms in the molecular geometry. If available, experimental geometries were taken

8


Table 1 Optimal exponents (a.u.−2 ) for H2 and D2 molecules with a 1s1p1d nuclear basisa . H2 D2 γs 23.680 34.797 γp 22.503 33.397 γd 20.284 32.054 a Calculations were performed with the APMO/HF method employing the Def2-TZVPD electronic basis set

from the NIST Computational Chemistry Comparison and Benchmark Database [63]. Otherwise, molecular geometries were optimized at the B3LYP/cc-pVTZ level of theory with the GAMESS computational package [64]. Mean polarizabilities α were calculated from the polarizability tensor α=

1 (αxx + αyy + αzz ). 3

(27)

PIEs were obtained as the difference of the mean polarizabilities of H (αH ) and D (αD ) isotopologues PIE = αH − αD . Additionally, we calculated the experimental PIE from PIEexp =

∆α αH . α0

(28) ∆α α0

data [17–20] as (29)

4 Results and Discussion 4.1 Nuclear basis set size effect on the PIE We calculated the polarizabilities of H2 O and D2 O employing a number of nuclear basis sets comprising different numbers of s,p and d-type GTFs. A regular electronic DFT calculation was performed for reference. Results presented in Figure 1 show that regardless of the nuclear basis set choice αH > αD > α∞ H . Figure 1 reveals that PIE values respond to the maximum angular momentum of the nuclear GTFs, but are practically insensitive to the number of GTFs with the same angular momentum. For instance, with 9s, 9s9p and 9s9p9d nuclear basis sets the H2 O molecule PIE is 0.0151 ˚ A3 , 0.0249 ˚ A3 and 0.0226˚ A3 respectively. The latter value is practically identical to the calculated with the 3s3p1d nuclear basis set, 0.0227 ˚ A3 . Therefore, all APMO/ADPT calculations reported in this work employed the 3s3p1d nuclear basis set.

4.2 APMO/ADPT polarizability isotope effects There are few PIE values reported in the literature [22]. We have selected 20 molecules from the data compiled in Refs [17–21] to benchmark our APMO/ADPT PIEs. We list in Table 2 experimental and theoretical αH and PIEs. As observed, APMO/ADPT αH ’s are on average 10 % larger than experimental values. Previous

9 0.030

1.64

0.025

1.60

H2O DO ∞ 2 H2O

1.58 1.56

0.020 0.015 0.010

1.54

0.005

1.52

0.000

(a)

1s 3s 5s 7s 9s 11s 13s 3s1p 3s3p 5s1p 5s3p 5s5p 9s5p 9s7p 9s9p 3s3p1d 3s3p3d 5s5p1d 5s5p3d 5s5p5d 9s9p5d 9s9p7d 9s9p9d

1.62

PIE / Å3

1.66

1s 3s 5s 7s 9s 11s 13s 3s1p 3s3p 5s1p 5s3p 5s5p 9s5p 9s7p 9s9p 3s3p1d 3s3p3d 5s5p1d 5s5p3d 5s5p5d 9s9p5d 9s9p7d 9s9p9d

Polarizability / Å3


(b)

Fig. 1 APMO/ADPT H2 O and D2 O polarizability (a) and PIE (b) calculated with different nuclear basis sets. ∞ H2 O denotes a regular DFT calculation with the BOA. Calculations were performed with the Def2-TZVPD electronic basis set.

studies [59, 60] have concluded that the LDA tends to overestimate αH by 5% for organic molecules. Figure S.1 in the supporting information reveals that there is a very good correlation (r = 0.999) between APMO/ADPT and experimental αH values. Further inspection of Table 2 reveals that on average APMO/ADPT PIEs almost double the experimental values. Nevertheless, as shown in Figure 2 there is excellent correlation (r = 0.985) between them. Therefore, semi-quantitative predictions of PIE values can be obtained by scaling APMO/ADPT values by a factor of 0.57. Table 2 shows that scaled APMO/ADPT PIEs values differ on average by 20% from experimental values. The PIE originates mainly from the contraction of the X-H bond upon deuteration [16]. Nakai and coworkers concluded that NMO methods overestimate nuclear quantum effects on bond distances, because the nuclear kinetic energy is contaminated with translational and rotational contributions [65]. For instance, the difference of the experimental distances of HCl and DCl is 0.0026 ˚ A[66], whereas the APMO calculated difference is 0.0053 ˚ A. Therefore, it is not surprising that the calculated PIE of HCl, 0.014 ˚ A3 , is higher than the experimental value, 0.006 ˚ A3 .

4.3 Polarizability isotope effects trends Previous studies for methanol and aniline revealed that experimental PIE are additive [19, 21] and that molecular PIEs can be approximated from bond polarizability isotope effect (BPIE) values [19]. Here, we test the additivity and bond type approximations to molecular PIEs using APMO/ADPT data. We calculated the PIE of a single H/D substitution in non-metals hydrides and used these values as representative BPIEs. Further analysis of the results presented in Table 3 reveals that as the electronegativity of X increases, the BPIE of a X-H bond decreases. Scaled values for O-H (0.0064 ˚ A3 ), N-H (0.0074 ˚ A3 ), and C-H 3 (0.0088 ˚ A ) bonds, are in close agreement with the experimental BPIEs reported by Van Hook and Wolfsberg [19] (0.0064 ˚ A3 , 0.0089 ˚ A3 and 0.010 ˚ A3 respectively).

10


Table 2 APMO/ADPTa and experimental αH and PIE (˚ A3 ). Molecule

isotopologues

Calculated Experimental Refs PIEb αH PIE [67, 19] hydrogen H2 / D2 0.033 ( 0.019 ) 0.819 d 0.011 f [68, 19] methane CH4 / CD4 0.061 ( 0.035 ) 2.593 d 0.042 f [67, 19] ammonia NH3 / ND3 0.041 ( 0.023 ) 2.220 d 0.026 f [69, 19] water H2 O / D 2 O 0.023 ( 0.013 ) 1.470 d 0.010 f [70, 19] hydrogen sulfide H 2 S / D2 S 0.037 ( 0.021 ) 3.782 d 0.016 f [67, 19] hydrogen chloride HCl / DCl 0.014 ( 0.008 ) 2.600 d 0.006 f [71, 17] hydrogen peroxide H 2 O2 / D2 O2 0.026 ( 0.015 ) 2.300 e 0.007 f [17, 20] chloroform CHCl3 / CDCl3 0.016 ( 0.009 ) 8.258 e 0.008 f [17] nitromethane CH3 NO2 / CD3 NO2 0.040 ( 0.023 ) 4.670 e 0.030 g [17] ethanol C2 H5 OH / C2 H5 OD 0.011 ( 0.006 ) 5.010 e 0.004 g [72, 19] ethylene glycol C2 H4 O2 H2 / C2 H4 O2 D2 0.028 ( 0.016 ) 5.610 e 0.013 f [17, 19] acetic acid CH3 CO2 H / CH3 CO2 D 0.014 ( 0.008 ) 5.038 e 0.009 f [17, 17] 1-propanolc C3 H7 OH / C3 H7 OD 0.013 ( 0.007 ) 6.805 e 0.005 g [17] glycerolc C3 H5 O3 H3 / C3 H5 O3 D3 0.040 ( 0.023 ) 7.751 e 0.021 g [73, 19] acetone (CH3 )2 CO / (CD3 )2 CO 0.093 ( 0.053 ) 6.390 e 0.059 f [17, 20] benzenec C6 H6 / C6 D6 0.097 ( 0.055 ) 9.950 e 0.054 f [17, 18] toluenec C6 H5 CH3 / C6 D5 CD3 0.131 ( 0.075 ) 11.861 e 0.065 f [17, 20] ciclohexanec C6 H12 / C6 D12 0.201 ( 0.115 ) 10.745 e 0.119 f [21] CH3 OH / CH3 OD 0.012 ( 0.007 ) 0.005 g [21] methanol CH3 OH / CD3 OH 3.575 0.047 ( 0.027 ) 3.265 e 0.030 g [21] CH3 OH / CD3 OD 0.058 ( 0.033 ) 0.034 g [17] C6 H5 NH2 / C6 H5 ND2 0.034 ( 0.019 ) 0.011 g [17] anilinec C6 H5 NH2 / C6 D3 H2 NH2 12.497 0.052 ( 0.030 ) 11.537 e 0.031 g [17] C6 H5 NH2 / C6 D3 H2 ND2 0.080 ( 0.046 ) 0.045 g Mean absolute percentage deviation: 10.0% 110.1% ( 19.8% ) a Calculations performed with the Def2-TZVPD:3s3p1d electronic:nuclear basis sets b Scaled PIE are presented in parenthesis (scale factor 0.57) c Exponents lower than 0.01 a.u.−2 were removed from the electronic basis set to achieve convergence in the SCF d Gas phase measurement e Liquid phase measurement f Experimental PIE estimated from reported ∆α data with Eq. 29 α0 g Experimental PIE estimated as the difference of reported α and α data D H αH 0.981 2.853 2.378 1.643 3.984 2.786 2.451 8.877 5.148 5.599 6.343 5.578 7.604 8.898 7.029 10.569 12.976 11.804

Molecular PIEs can be approximately calculated from BPIE values employing the following equation PIE ≈

X

NXH/D BPIEXH ,

(30)

X

where NXH/D is the number of H atoms replaced with D of a particular bond type and BPIEXH is the value reported in Table 3. We assess the accuracy of the additivity approximation calculating the PIEs of 100 pairs of isotopologues using Eq. 30 and BPIEs of Table 3 and contrasting them with PIEs calculated at the APMO/ADPT level. We list in Table S.1. in the supporting information the set of molecules employed with their estimated and calculated PIEs. Figure 3 presents a correlation plot of these PIEs. The correlation coefficient of r = 0.989 confirms that the BPIE are approximately additive. However, further analysis of the differences between the PIEs estimated by using Eq. 30 and by APMO/ADPT in Table S.1 shows a mean absolute deviation is


11

Fig. 2 Comparison of APMO/ADPT and experimental PIEs.

Experimental PIE / Å3

0.12 0.09 0.06 0.03 0.00 0.00

y=0.57x r=0.985 0.05

0.10

0.15

0.20

Calculated PIE / Å3 Table 3 APMO/ADPT Bond polarizability isotope effects (˚ A3 ) a Bond Isotopologues BPIEXH H-H H2 /HD 0.017 (0.010 ) B-H BH3 /BH2 D 0.018 (0.010 ) C-H CH4 /CH3 D 0.015 (0.0088) N-H NH3 /NH2 D 0.013 (0.0074) O-H H2 O/HDO 0.011 (0.0064) F-H HF/DF 0.0081 (0.0046) Si-H SiH4 /SiH3 D 0.026 (0.015 ) P-H PH3 /PH2 D 0.020 (0.011 ) S-H H2 S/HDS 0.019 (0.011 ) Cl-H HCl/DCl 0.014 (0.0081) a Calculations performed with the Def2-TZVPD:3s3p1d electronic:nuclear basis sets. Scaled PIE are presented in parenthesis (scale factor 0.57)

11%. Therefore, we conclude that the additivity approximation employing BPIEs only allows for semi-quantitative predictions of molecular PIEs.

5 Conclusions We proposed a methodology that combines the APMO and ADPT approaches to calculate molecular PIEs. We assessed the performance of the APMO/ADPT methodology contrasting the calculated PIE for 20 molecules with experimental data. Our APMO/ADPT results present a good correlation with experimental PIEs. However, APMO/ADPT PIEs are always overestimated, because the APMO method magnifies the changes in bond length upon deuteration. Scaling APMO/ADPT results by a factor of 0.57 allows PIE predictions with an average error of 20%. We have calculated the PIE of non-metals hydrides isotopologues with a single H/D substitution. We have found that the PIE of these X-H bonds decrease by

12


Fig. 3 Comparison of PIEs estimated with equation 30 and calculated with APMO/ADPT

Calculated PIE / Å3

0.20 0.15 0.10 0.05 0.00 0.00

y=x r=0.989 0.05

0.10

0.15

0.20

PIE estimate from BPIE / Å3

increasing electronegativity of X. Molecular PIEs can be approximately estimated by adding representative bond PIE. On average, estimated values differ by 11% from the regular APMO/ADPT PIEs. Errors in APMO/ADPT PIEs are mainly associated to the error in the geometric isotope effect obtained with NMO methodologies. This errors can be reduced by the inclusion of nuclear-electron correlation [74]. We are currently working on methodologies to efficiently include nuclear-electron correlation in APMO calculations.

6 Supporting Information Figure S.1. contrasts calculated and experimental polarizabilities reported in Table 2. Table S.1. presents the set of 100 isotopologues pairs employed to test the accuracy of the additivity approximation. Acknowledgements We gratefully acknowledge the support of Universidad Nacional de Colombia and Universidad de la Amazonia.

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Theoretical Investigation of Polarizability Isotope Effects F´elix Moncada, Roberto Flores-Moreno, Andr´es Reyes September 29, 2016

1

Supporting Information

Experimental αH/ Å3

12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

Calculated αH / Å3 Figure S.1. APMO/ADFT calculated polarizabilites and experimental polarizabilites of the molecules listed in Table 2. Solid line represents y=0.91x. R2 = 0.998. Calculations performed with the Def2-TZVPD:3s3p1d electronic:nuclear basis sets.

1

Table S.1. APMO/ADFTa calculated αH , αD and PIE. BPIE estimates with Eq. 30. All values in ˚ A3 . molecule

isotopomers

1-propanol 2-propanol acetic acid acetic acid acetone acetylene acetylene ammonia ammonia aniline aniline aniline benzene borane borane borazine butane chloramine chloroacetylene chloroform chlorosilane ciclohexane cyanamide diazene dichlorosilane difluoroborane dimethylborane dimethylborane dimethylsilane diphosphane disilane ethane ethanethiol ethanethiol ethanol ethene ethene ethylenglicol fluoramine fluorosilane formaldehyde formic acid formic acid formic acid glycerol HPO hydrazine hydrogen hydrogen azide hydrogen boron sulfide hydrogen cyanide hydrogen disulfide hydrogen disulfide

C3 H7 OH/C3 H7 OD (CH3 )2 CHOH/(CH3 )2 CHOD CH3 COOH/CD3 COOH CH3 COOH/CH3 COOD CH3 COCH3 /CD3 COCD3 C2 H2 /C2 D2 C2 H2 /C2 HD NH3 /ND3 NH3 /NHD2 C6 H5 NH2 /C6 D3 H2 ND2 C6 H5 NH2 /C6 H5 ND2 C6 H5 NH2 .out/C6 D3 H2 NH2 C6 H6 /C6 D6 BH3 /BD3 BH3 /BHD2 B3 H3 N3 H3 /B3 D3 N3 H3 C4 H10 /C4 D10 NH2 Cl/ND2 Cl C2 ClH/C2 ClD CHCl3 /CDCl3 H3 SiCl/D3 SiCl C6 H12 /C6 D12 NH2 CN/ND2 CN N2 H2 /N2 D2 H2 SiCl2 /D2 SiCl2 BF2 H/BF2 D (CH3 )2 BH/(CD3 )2 BD (CH3 )2 BH/(CH3 )2 BD (CH3 )2 SiH2 /(CH3 )2 SiD2 P2 H4 /P2 D4 Si2 H6 /Si2 D6 C2 H6 /C2 D6 C2 H5 SH/C2 D5 SD C2 H5 SH/C2 H5 SD C2 H5 OH/C2 H5 OD CH2 CH2 /CD2 CD2 CH2 CH2 /CH2 CD2 C2 H4 O2 H2 /C2 H4 O2 D2 NH2 F/ND2 F H3 SiF/D3 SiF CH2 O/CD2 O HCOOH/DCOOD HCOOH/DCOOH HCOOH/HCOOD C3 H5 O3 H3 /C3 H5 O3 D3 HPO/DPO N2 H4 /N2 D4 H2 /D2 N3 H/N3 D BSH/BSD HCN/DCN H2 S2 /D2 S2 H2 S2 /HDS2 2

APMO/ADFTa αH αD PIE 7.604 7.591 0.013 7.689 7.674 0.015 5.578 5.532 0.045 5.578 5.564 0.014 7.029 6.936 0.093 3.604 3.581 0.024 3.604 3.592 0.012 2.378 2.337 0.041 2.378 2.354 0.025 12.497 12.417 0.080 12.497 12.463 0.034 12.497 12.445 0.052 10.569 10.472 0.097 2.955 2.899 0.055 2.955 2.918 0.036 9.818 9.763 0.056 8.884 8.729 0.155 4.289 4.263 0.026 5.079 5.067 0.012 8.877 8.862 0.016 6.882 6.810 0.072 11.804 11.603 0.201 4.195 4.172 0.023 3.000 2.960 0.039 8.571 8.528 0.044 2.700 2.680 0.020 7.141 7.010 0.130 7.141 7.117 0.024 9.180 9.131 0.049 9.338 9.264 0.073 10.216 10.062 0.154 4.816 4.724 0.092 8.019 7.927 0.092 8.019 8.004 0.016 5.599 5.587 0.011 4.450 4.404 0.046 4.450 4.423 0.027 6.343 6.314 0.028 2.430 2.401 0.029 4.746 4.687 0.060 2.935 2.900 0.035 3.680 3.648 0.032 3.680 3.663 0.018 3.680 3.670 0.011 8.898 8.858 0.040 4.489 4.463 0.026 3.775 3.717 0.058 0.981 0.948 0.033 3.862 3.846 0.015 5.297 5.272 0.025 2.625 2.613 0.012 7.043 6.994 0.049 7.043 7.017 0.026

Bond PIEb 0.011 0.011 0.046 0.011 0.093 0.031 0.015 0.039 0.026 0.072 0.026 0.046 0.093 0.053 0.035 0.053 0.154 0.026 0.015 0.015 0.077 0.185 0.026 0.026 0.052 0.018 0.110 0.018 0.052 0.080 0.155 0.093 0.096 0.019 0.011 0.062 0.031 0.022 0.026 0.077 0.031 0.027 0.015 0.011 0.034 0.020 0.052 0.034 0.013 0.018 0.015 0.038 0.019

hydrogen peroxide hydrogen sulfide hydrogen trisulfide hydrogen trisulfide hydroxylamine hydroxylamine hydroxylamine hypofluorous acid hypophosphorous acid hypophosphorous acid methane methanethiol methanethiol methanol methanol methanol methylamine methylamine methylborane methylborane methylphosphine methylphosphine methylphosphine methylsilane methylsilane methylsilane nitromethane nitroxyl pentane phosphine phosphine phosphine oxide phosphinous acid phosphinous acid phosphorous acid phosphorous acid propane propyne silane silane silane silene thiohydroxylamine thiohydroxylamine toluene trichlorosilane water

b

H2 O2 /D2 O2 H2 S/D2 S H2 S3 /D2 S3 H2 S3 /HDS3 NH2 OH/ND2 OD NH2 OH/ND2 OH NH2 OH/NH2 OD HOF/DOF H2 POOH/D2 POOD H2 POOH/D2 POOH CH4 /CD4 CH3 SH/CD3 SD CH3 SH/CH3 SD CH3 OH/CD3 OD CH3 OH/CD3 OH CH3 OH/CH3 OD CH3 NH2 /CD3 ND2 CH3 NH2 /CH3 ND2 CH3 BH2 /CD3 BD2 CH3 BH2 /CH3 BD2 CH3 PH2 /CD3 PD2 CH3 PH2 /CD3 PH2 CH3 PH2 /CH3 PD2 CH3 SiH3 /CD3 SiD3 CH3 SiH3 /CD3 SiH3 CH3 SiH3 /CH3 SiD3 CH3 NO2 /CD3 NO2 HNO/DNO C5 H12 /C5 D12 PH3 /PD3 PH3 /PHD2 H3 PO/D3 PO H2 POH/D2 POD H2 POH/D2 POH HPO(OH)2 /DPO(OH)2 HPO(OH)2 /HPO(OH)2 C3 H8 /C3 D8 HCCCH3 /DCCCH3 SiH4 /SiD4 SiH4 /SiH2 D2 SiH4 /SiHD3 CH2 SiH2 /CH2 SiD2 NH2 SH/ND2 SD NH2 SH/NH2 SD C6 H5 CH3 /C6 D5 CD3 HSiCl3 /DSiCl3 H2 O/D2 O

2.451 3.984 10.894 10.894 3.110 3.110 3.110 1.846 5.347 5.347 2.853 5.983 5.983 3.575 3.575 3.575 4.330 4.330 5.007 5.007 6.945 6.945 6.945 7.240 7.240 7.240 5.148 2.427 10.945 4.984 4.984 4.955 5.442 5.442 5.768 5.768 6.868 5.526 5.354 5.354 5.354 7.517 5.377 5.377 12.976 10.255 1.643

2.425 3.947 10.862 10.878 3.071 3.081 3.096 1.835 5.294 5.306 2.792 5.915 5.969 3.517 3.528 3.563 4.253 4.303 4.914 4.966 6.856 6.897 6.905 7.122 7.189 7.162 5.109 2.406 10.746 4.923 4.949 4.896 5.393 5.398 5.752 5.726 6.738 5.512 5.253 5.302 5.278 7.476 5.331 5.358 12.845 10.227 1.621

a Calculations performed with the Def2-TZVPD:3s3p1d electronic:nuclear basis sets PIE calculated with Eq. 30 and the BPIE values presented in Table 3

3

0.026 0.037 0.032 0.016 0.040 0.029 0.015 0.011 0.053 0.041 0.061 0.068 0.014 0.058 0.047 0.012 0.077 0.026 0.093 0.042 0.089 0.048 0.040 0.118 0.051 0.078 0.040 0.021 0.200 0.061 0.035 0.059 0.050 0.044 0.016 0.042 0.130 0.013 0.100 0.052 0.076 0.041 0.046 0.020 0.131 0.028 0.023

0.022 0.038 0.038 0.019 0.037 0.026 0.011 0.011 0.051 0.040 0.062 0.065 0.019 0.057 0.046 0.011 0.072 0.026 0.082 0.035 0.086 0.046 0.040 0.124 0.046 0.077 0.046 0.013 0.185 0.060 0.040 0.060 0.051 0.040 0.020 0.042 0.123 0.015 0.103 0.052 0.077 0.052 0.045 0.019 0.123 0.026 0.022