Electric Permittivity in Individual Atomic and ... - ACS Publications

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Electric Permittivity in Individual Atomic and Molecular Systems Through Direct Associations with Electric Dipole Polarizability and Chemical Hardness Paweł Szarek* Center for New Technologies, University of Warsaw, Ż wirki i Wigury 93, 02-089 Warsaw, Poland S Supporting Information *

ABSTRACT: The particular role in the interaction with external electromagnetic fields is played by local dielectric environment, hence the electric permittivity of a medium. This includes both the intramolecular screening effects as well the neighbor interactions with, i.e., solvent molecules, which conclude in modified electronic transport properties, i.e., of single-molecule junctions. Specific features of molecular (a few electron) systems require a unique approach to their characterization because transfer of just one electron alters the system significantly. Under these circumstances the electron−electron interaction depicted through a screening phenomenon in individual systems has been directly related to electric dipole polarizability and chemical hardness. The relation with these two fundamental electronic response functions quantifies a physical basis for field energy storage as associated with charge delocalization. This sheds new light on how the chemical reactivity affects behavior of molecular electronic devices. Moreover it might also serve to assess how electromagnetic fields can alter the reactivity or assist in catalysis.

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interaction with electromagnetic fields and photons or charge separation and transport. In this context, elementary techniques of manipulation of individual, precisely tailored molecular devices could be considered to share many analogies with methods used in telecommunication systems. Molecular devices, likewise specially designed antennas, might serve as receivers and transmitters of signals (wave packets) in selected bands of frequencies, which is intimately connected with electric permittivity. This work is a first attempt to examine the nature of dielectric properties in atoms and single-molecule systems in reference to their electronic response functions. Although the model presented is not a first-principles theory, it reveals important ties between electromagnetic fields and the reactivity, using first-principle observables. It allows exploring the impact of the chemical nature and quantum effects on engineering of molecular electronic components. It also suggests how particular properties of atoms or molecules might be directly incorporated into functionality of future molecular-based technologies, which will make possible utilizing the full potential of the new frontier. Moreover the leading mechanisms and working principles of molecular devices are indicated, and possibilities of electromagnetic field assistance in catalysis are discussed.

INTRODUCTION Dielectric materials can be thought of as composed of dielectric atoms or molecules distributed throughout vacuum space. Hence the local field acting on each entity is higher than the measured averaged field in a material. The observed macroscopic dielectric properties are also affected through the polarization field generated by the surrounding species. Moreover electric field related charge deformation effects lead to changes in density of materials. In consequence singlemolecule electronic devices are fundamentally different from complementary metal-oxide semiconductor (CMOS) technology. This holds true even if fabricated in nanodimensions or the single electron limit.1−8 The difference in behavior between a single molecule and an aggregate or between a dilute and condensed phase constitutes an immensely important problem. The collective effects are often observed while going from single atoms, through molecules to larger aggregates, or one-, two-, or three-dimensional materials (for example, changing properties in various size metal clusters). On the contrary, individual molecules usually have a moderate number of electrons, and even fewer are directly engaged in their physicalchemical activity and exhibit peculiar features critically dependent on thermal and electrostatic effects of the local surroundings. An important aspect of development of molecular systems is associated with control of molecular processes. Especially essential are methods of their external manipulation with electromagnetic fields.9 In particular the local dielectric properties of a molecular junction play a crucial role in the © XXXX American Chemical Society

Received: March 20, 2017 Revised: May 18, 2017 Published: May 19, 2017 A

DOI: 10.1021/acs.jpcc.7b02626 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C The key processes in molecular electronics scale down to a regime where quantum effects are of the most importance, while the transfer of one electron alters the system significantly. Quantum mechanics offers a fundamental description of matter, with embedded effects of electric and magnetic fields on its basic elements, and enables properties such as polarizability, Fermi level, or energy gap to be calculated from the first principles. Customarily the charge transport properties in molecular junctions for the most part can be traced back to the difference between the donor and acceptor energy levels (energy gap) of the molecular bridge.10 Nevertheless those predictions are not universal (since many other factors are involved such as inelastic effects, bridge length, and coupling with electrode states) and sometimes fail to provide correct description of conductance.11,12 The alternative predictions of conductance in molecular junctions can also be traced to molecular polarizability13−16 or electric dipole moment.17 In former models the local dielectric properties of a junction governed by molecular bridge are responsible for the device current−voltage characteristic and charge transport mechanism. This identifies the primary electronic response functions, along with electric permittivity, as key features determining charge transport properties of molecular electronic systems. Although those descriptors contain crucial information about junction conducting behavior, none by itself draws a complete picture or can be uniquely applied to its characterization. Therefore, it is imperative to understand how they overlap and link with each other, which in consequence will let us describe molecular systems more accurately.

Apart from the relations between electric dipole polarizability and hardness the obvious associations can be found of polarizability with dielectric constant (Clausius−Mossotti relation, through dipole moment,28 and locally with the use of dielectric polarization density). Likewise, numerous associations exist of hardness to permittivity29,30 (or refractive index31−36 in semiconductors, such as Penn,29 Ravindra,37 and Moss38 relations, or molar refractivity39). These suggest that the interrelation directly linking together electric dipole polarizability, hardness, and electric permittivity could be identified. Such dependency could allow exploring new dimensions of device operability that are exclusively accessible at the molecular level and linked to reactivity and quantum phenomena. Moreover, it can bring an understanding of how the chemical processes can affect the behavior of corresponding molecular electronic devices or how electromagnetic fields can alter the reactivity and might assist in catalysis. Direct Approach to Permittivity in Atomic and Molecular Systems. The dimensions of atoms in their ground state are strictly related to hardness or polarizability, while radii of successive electronic shells have an impact on the effectiveness of screening. The simple expression presented in eq 1 without ambiguity assigns the radius to any entity even in ionized or excited states (although the boundary of an atom or molecule in general is not a well-defined physical property). The definition can also be employed, i.e., in coarse-grid models40,41 to estimate sizes of beads representing not only atoms but also molecules or nanoparticles as well. This is because it can be interpreted in terms of standard deviation of the Fukui function spread out around the mean23,42

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THEORY The chemical hardness18 (band gap analog) and the electric dipole polarizability are two primary physical entities that are associated with electronic responses of materials to external influences. The accompanying (complementary in a way) permittivity carries essential information on medium interaction with electromagnetic fields. At heart it characterizes the electric field energy storage in a material and becomes a manifestation of the electron−electron interaction through a screening phenomenon. Basic Links between Key Electronic Response Functions. Numerous relations link hardness and polarizability. Beginning with a general and simple concept, they are loosely coupled through HSAB theory,19 which discriminates electron donor/acceptor behavior of “hard”, weakly polarizable species against strongly polarizable “soft” ones. They also associate through the minimum polarizability principle,20 which is a reformulation of the maximum hardness principle.21 A more straightforward relationship is made through local density approximation of the electronic polarizability,22 matching it with the inverse of hardness. Recently, on the basis of the former two principles, it has been shown that the product of the chemical hardness and polarizability can be used to estimate the atomic radii23 R = e−1 αη

R∼

∫ r 2f (r)dr − (∫ rf (r)dr)2

(2)

Additionally, in the case of atoms it determines distance to the maximum radial probability of the electron density of the highest occupied orbital.23 Furthermore, adopting such radial distribution of electronic density should minimize the system energy.23 Employing atomic radii in combination with classical concepts (the correspondence principle43−46) allows approximating selected physical and chemical properties.47−51 In particular, the atomic capacitance can be estimated using a simple model of a hallow conducting sphere with size determined by atomic radius52 C = 4πε0R

(3)

Likewise the isotropic electric dipole polarizability is obtained α = 4πε0R3

(4) 23,53−61

Among various nonequivalent definitions of atomic radii if the Bohr radius, a0, is chosen in the case of the simplest, a hydrogen atom, the above model is found to yield polarizability approximately 4.5 times smaller than the experimental value. The error of comparable size arises for capacitance as well. Accordingly such a sphere radius used in the model may be inadequate, nearly R = 1.5a0; hence the average distance of an electron from the nucleus in a hydrogen atom, or the R = 4a0, yields more accurate results in the case of polarizability and capacitance, respectively. Yet these are divergent values. The other radii perform poorly in a similar manner. Therefore, permittivity of such a sphere should be considered other than vacuum permittivity. The latter scenario in general is a

(1)

Alternatively, few other relationships with polarizability can be made on the basis of ionization potential and oscillator strength.24−27 It is also possible to establish correlation through conductance since both the hardness and the polarizability have been used to characterize this property in molecular junctions.10−17 B


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The Journal of Physical Chemistry C

of eq 10, compared to 4.11 for eq 6. Following the result of eq 10 the interpretation of dielectric constant in terms of moments in probability theory is possible. It yields a combination of measure of asymmetry of HOO density function (3rd moment - skewness, squared), and its spread around the mean (the variance, second moment in cubic power). Accordingly the asymmetric distributions of HOO density would be associated with higher permittivity and more spatially spread HOO charge distributions with lower values. It can be understood in terms of the attractive force between valence electric charges and the nucleus in atoms, causing deformation within the volume of the “dielectric sphere”. Effect size correlates with permittivity magnitude and takes place in order to reduce the stored potential energy of the system. The associated compressive pressure results proportionally to nuclear field intensity. Although eq 10 by itself is a poor approximation considering empirical fitting constants, it indicates a direct relation of dielectric constant with charge density distribution. However, any threshold or rule of thumb here is arbitrary since many atomic distributions might exhibit similar moments. Nevertheless it can be attempted to propose a direct estimate related to charge density distribution (over the dependence on Fukui function via R, i.e., reformulated eqs 5 or 7). Applying the delta method74 to reorganized capacitance formula will produce

manifestation of electron−electron interactions in the form of a simple yet important screening phenomenon.62−64 Since one electron variation in atoms or moderate size molecules leads to significant repercussions, their capacitance is expressed by the quantum capacitance. The capacitance associated with charging of an atomic or a molecular system is identified with the work of single-electron transfer with respect to the vacuum energy level, thus with ionization potential and electron affinity.65,66 Therefore, we assume C=

e2 = 4πεR ε0R η

(5)

and substituting the radius defined in eq 1, we find 4πεR ε0 =

e3 αη3

(6)

From this point forward all formulas will be presented in atomic units, thus omitting some unit constants, in order to highlight important relations between different key properties. Hence the polarizability is [au]

α = 4πεR ε0R3 = ⇒ εR R3

(7)

Equation 7 is now an identity relation considering the permittivity could be determined using eq 1 and either eq 5 or eq 7 or by combining together the latter two. In light of eq 5 the electrostatic character of hardness,67,68 expressed via the Coulomb integral defined by the charge density of frontier orbitals (Fukui function) and hardness kernel,69−71 has to be emphasized. The product of such an electric potential energy (due to a unit charge distribution) with the expected value of R is naturally proportional to the dielectric constant of a material. This is a classical screened Coulomb interaction since it includes the one-electron interactions with background nuclear charge shielded by the remaining electrons, which is conforming to the Hohenberg−Kohn theorem.72 Yet it is important to point out that such a definition does not include the nuclear repulsion because the external potentialthe hardness kernelis constant.69−71 Uniformly Charged Sphere vs Local Response of Charge Density. Reformulation of eq 6 in terms of approximations referring to charge density of the highest occupied orbital (HOO, atomic or molecular)73 provides an additional prospective α = kα

∫ ρHOO(r)r 3dr = kα⟨r 3⟩HOO

εR =

∫

⟨r ⟩f

⇒ εR ≅

2

1 η

∫

f (r ) dr r

f (r ) dr ≡ =f = −(Vel(r0)N + 1 − Vel(r0)N − 1)/2 r

ηD(r ) =

(8)

(

dVel(r ) dN

(11)

(12)

)

v

ρ(r ) , N

can be used to assess the =f (eq 13,

in molecules the total ρ(r) might be replaced by atomic density fractions, according to, i.e., Hiershfeld scheme, eq 14). (9)

=f =

∫ ηD(r)dr = ∫

=f =

∫

which yields an elegant relation 2 3 ⎛ ⎞−1/2 3 ⟨r ⟩HOO εR = ⎜kαk η 3 2 ⎟ ⟨r ⟩HOO ⎠ ⎝

2

Moreover the simple screening model in an isolated atom 1 indicates a linear relation with hardness, η = 4 ⟨r −1⟩f .79 However, the Fukui potential at nuclei might not constitute a good local definition of hardness of atoms in molecules,80 and its failures have been pointed out.81 It is essential to mention that formally the =f is the hardness derivative, as follows from the local hardness approximation h(r) = −Vel(r)/2N.82 Moreover, the integral of local hardness, defined83 as

⟨r 2⟩HOO ⟨r 3⟩HOO

2

where ⟨r2⟩f is the second moment (variance) of the Fukui function. In addition the Fukui potential (also known as hardness potential69) defined for charge distribution that matches the Fukui function, in the case of atoms, might be estimated as a difference in electrostatic potential from electrons at the atomic nucleus76−78

and η = kη

11 1 ≅ ηR η

⎛ = ⎜⎜ ⎝

(10)

with empirical parameters kα = 1.48 and kη = 0.50, determined for a large set of elements.73 However, the best estimates for hydrogen atom are kα ≅ 0.57 and kη ≅ 0.59, and corresponding values of integrals, ⟨r3⟩HOO = 7.5a03 and ⟨r2⟩HOO = 3a02.73 Therefore, the εR estimated for the hydrogen atom is found to be 4.22 (or 4.17 numerically, B3LYP/6-311++G**), in the case

∫

ΔVel(r ) ρ(r ) dr ΔN N

(13)

f (r ) dr r ⎞⎛ ⎞−1 ⎛ dVel(r ) ⎞ ρi ρi (r )dr ⎟⎟⎜⎜ ρ(r ) (r )dr ⎟⎟ ⎜ ⎟ ρ (r ) ⎝ dN ⎠v ρpro ρpro ⎠⎝ ⎠

∫

(14)

Electric Field Role in Reactivity. Subsequent to the presented relationships with local response of charge density, the εR might be recognized as a quotient of an average Fukui C


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Figure 1. Single atom permittivity estimated by eq 6 (red squares). The hardness has been calculated with electron affinity, EA ≥ 0, and the empty squares show corresponding solutions with negative electron affinity, EA < 0. The permittivities obtained with the Clausius−Mossotti relation, eq 17, using consistent volumes are marked with circles. The numerical estimates by Fukui potential, according to eq 11 (gray diamonds) for H to Kr atoms, at B3LYP/6-311++G** level of theory are also given.

tied to the dielectric constant of a body). In principle this expression has been resolved to describe the dielectric sphere87 (the condition that has been imposed on a conducting sphere model through eqs 5 and 7)

potential (hardness potential, which is considered an electrostatic part of hardness) to global hardness. In this context, the trends observed in reactive sites, characterized by charge transfer and determined with Fukui potentials,75,84 are more evident as related to the process of storing the electric field energy. In other words, it describes the overall electric flux generated in molecular sites by unit charge (decrease in electric field between charges). Hereby the reagent is guided toward a site where it can accept or donate an electron with respect to its electrophilic or nucleophilic nature. The values of =f were correlated with intramolecular and intermolecular reactivity trends, matching the experimental reaction rates.75,77,85,86 However, the polarization process in the target reagent induced by the approaching electrophile/nucleophile is not taken into account in the =f index.75,77,80,84−86 The response of the system to changing electron density, upon reagents coming into contact, however has been incorporated within permittivity. Thus, the molecules or atoms with large permittivities are drawn from the weak field to intense field region, while those with lower permittivities are displaced from intense fields into weak field regions in the same fashion (recall the Table of Contents Graphics). The associations with Fukui potential lead to an additional two relations R2 ≅ α≅

1 1 ⟨1/r ⟩f −2 = =f −2 ∼ ⟨r 2⟩f 2 2 11 ⟨1/r ⟩f −2 = (2η =f 2)−1 2η

N

α⎡ εR − 1 α ⎤ = V ⎢ ≡ 3⎥ εR + 2 3ε0 ⎣ R′ ⎦

(17)

N represents number of atoms per volume, V. In the case of a single atom, N = 1, the reasonable assumption for a volume would be a sphere defined by a radius. However, the Clausius− Mossotti formula, based on the Lorentz local field, has been derived for the lattice of dipoles with cubic symmetry. In such a case R′ = 2R, which corresponds to the cell diameter. The value of εR, eq 17, for the hydrogen atom is found to be 4.07. The discrepancy regarding the use of R in eqs 5 or 7 and R′ = 2R in the Clausius−Mossotti model, eq 17, originates from distinctive boundary conditions in both models and the definition of R itself. A hallow conducting sphere model (eq 3) has been modified into a uniformly charged sphere (eq 5, linearly polarizable in every volume element), with the nucleus point charge at the center screened by electron shells. As a consequence, the R representing the greatest radial probability of finding the highest occupied (“outer most”) electron is related to the expected volume of screening charge density of inner shells. Moreover the polarizability might be represented as the square of a dipole moment over the excitation energy, assuming that the mean excitation energy corresponds to removal of an electron from distance R, thus from the HOO to infinity (E = e2/4πεrR). Consequently the most important transition dipole moment is approximately equal to the charge of an electron multiplied by R, which yields α = e2R2[e2/ 4πε0εRR]. On the other hand the Clausius−Mossotti model is found on the cavity related to the entire physical volume of the polarizable dielectric body, with the point dipole at the center. The fundamental difference between these two types of polarizations is that in the dielectric sphere every volume element is polarized, while in a conducting sphere the induced dipole moment arises from surface charges. The square of the dielectric constant might be regarded as a proportionality parameter between the cubic softness (inverse of hardness cube) and polarizability. Therefore, it quantifies the principle associated with attributing low polarizability to hard

(15)

(16)

which strengthen the choice of R in the role of atomic radius and draw a direct connection between polarization phenomena and energy required to promote a valence electron bound to an atom, to unoccupied levels. Simultaneously the energy gap can be thought of as a barrier which prevents charge transfer (electron−hole pair generation) and in consequence the charge recombination (density flow) in the system that leads to chemical reaction or electrical conductivity. Classical Dielectric Sphere. The dielectric constant obtained with eq 6 can be confronted with the Clausius− Mossotti formula (where the ratio of an average isotropic polarizability to a volume occupied by the dielectric medium is D


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where Vi = 3Vm/NA for medium organized in regular lattice, or Vi = 3Vm/4πNA in the case of amorphous/disordered medium (actually in the latter, the Vi = (2l ̅)3 is related to mean free path; however, the simple relation with molar volume might be an acceptable approximation). The 1 stands for relative permittivity of vacuum. Figure 2 (or Figure S1 in Supporting

systems. The size of the energy gap (i.e., HOMO−LUMO) reflects the energy associated with delocalization of charge. If it is low, it is easier for the electron to “hop” between orbitals and thus more susceptible to spatial delocalization, which means higher polarizability. Conversely, large gaps indicate a greater barrier for electron delocalization and lower polarizability. Moreover the link between electric dipole moment and the minus derivative of energy of a system with respect to the field, arising out of the Hellman−Feynman theorem (by the means of Stark energy expression), underlines the role of permittivity (screening) in coupling between the induced dipole moment and the energy gap.

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RESULTS The single particle permittivity is defined in the limit of a/v1/3 → 1, where a is the characteristic diameter of the particle and v1/3 is the interparticle extent (correspondingly to volume held). Under such conditions the permittivity evolves into εR = (αη3)−1/2. Figure 1 (and Table S1 in Supporting Information) presents a collection of permittivities calculated using experimental data by eqs 6 and 17 or estimated numerically with eq 11 (at B3LYP/6-311++G**), for a number of atoms up to atomic number 87. Three estimates correlate nicely with each other. The elements, of which the negative ions are metastable, thus exhibit negative electron affinity energies:88 He, Be, N, Ar, Mn, and Xe, are presented including the results with EA = 0 presumed. However, the values of radii determined with EA < 0 are greater and the permittivities lower. Yet, it is unclear which of the low-lying metastable states should be considered and what type of corrections to polarizability are introduced. In particular Ar with EA = 0 has permittivity εR = 1.90, although the state with EA = −11.5 eV and 260 ns lifetime88 (and invariant polarizability) yields εR = 0.84 according to eq 6, which is obviously wrong. Respective radii are 1.80 au vs 2.36 au for the metastable state, which when inserted into eq 17 returns an acceptable value of εR = 1.35. Generally the s2 and half-filled p3, d5 or fully filled p6, d10 electronic configurations exhibit lower permittivities than other shell occupancies. Overall the highest atomic permittivities are achieved in systems with partially filled p orbitals. Relation to Empirical Dielectric Constant. The generic dielectric properties can be obtained by adopting characteristic volume per particle (interparticle extent) as related to the molar volume over the Avogadro number, Vm/NA, with the exact values of polarizabilities. This recovers the empirical permittivity of a medium composed of nonpolar particle gas (the additional constraints include linear polarization and lack of interaction). In such a case accounted for, the Lorentz formula with the local field correction (eq 18, equivalent to eq 17) is expressed as −1 NA ⎛ NA ⎞ εr − 1 = α ⎜1 − α ⎟ Vm ⎝ 3Vm ⎠

Figure 2. Empirical estimates of dielectric constant of atoms based on molar volume, eq 18, compared to effective permittivity obtained by averaging dielectric and vacuum volume fractions of a medium, according to eq 19. Refer also to Figure S1 in the Supporting Information.

Information) compares empirical and effective permittivity of atoms. A sound correlation is found between individual atomic permittivity and macroscopic empirical dielectric constant. Additionally Table 1 summarizes values of permittivity for rare gas atoms, confirming established relations. Permittivity in Polyatomic Species. The radius and permittivity defined through eq 1 and eq 6 might be also applied to molecules, assigning some idealized volume and permittivity. Table 2 lists estimated values for a few exemplary compounds. Although the individual atomic permittivities do not sum up to the total molecular value, it is possible to approximate the respective effective susceptibility of a chemical compound through the volume-weighted sum of individual atomic susceptibilities of isolated constituents, with empirical scaling parameter, kX ∼ 0.5, and the vacuum permittivity assigned to excess molecular volume (Table 2). εR m ≅ 1 +

(18)

εr ≅

Rm ≅

∑ R i2 i

(20)

=

∑ αiηi

(21)

i

or with better agreement by 1 R m ≅ ((∑ R i 2)1/2 + (∑ R i 3)1/3 ) 2 i i

3

(Vi − (2R ) ) ·1 + εR (2R ) Vi

Vmol

If molecular volume is unknown it might be estimated using the effective radius derived from atomic radii of constituents

This empirical value, εr, of permittivity differs from εR on account of interparticle extent/“designated volume” to an individual particle. Through averaging permittivities over volume fractions in medium, of dielectric and vacuum domains, the effective permittivity of a medium is obtained 3

kX ∑ Vat i(εRi − 1)

(19) E

(22)


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Table 1. Empirical Dielectric Constant of Rare Gases in Different Phases,89−91 Calculated Permittivity, εR, in Isolated Atoms, and Effective Permittivitya He gas liquid solid (eq 6) (eq 17) (eq 19): gas liquid solid

Ne

Ar

Kr

Xe − 1.874 2.033 1.776 1.856

b

1.0000684 1.055 − 1.150c 1.151c

1.00013 1.53 − 2.418 2.299

1.000516 1.504 1.599 1.901 1.935

1.000768 1.657 1.784 1.828 1.888

1.000024 1.018 ± 0.002 −

1.00021 1.285 ± 0.007 1.335

1.00070 1.552 ± 0.006 1.637

1.00103 1.672 ± 0.007 1.780

1.00161 1.842 ± 0.005 1.971

a The liquid-state effective permittivities are averaged over molar volumes for melting and boiling points. bRef 92. cWith electron affinity EA = −19.7 eV; ref 88.

Table 2. Calculated Permittivity of Selected Moleculesa N2 CH4 H2O H2O theoret. (H2O)2 theoret. CO ″ CO2 ″ CH2O A (v.) A (a.) G (v.) G (a.) C (v.) C (a.) T (v.) T (a.)

IP (eV)

EA (eV)

α (au)

η (au)

Rmol

Vmol

εR

εr

εrvol. av.

b

−1.4 1.2e −0.17f −0.80 −0.62 1.326b −1.6b −0.600b −1.60b −0.65g −0.54i 0.012l 0 0 −0.32i 0.085m −0.29i 0.068l

d

0.3120 0.2097 0.2350 0.2698 0.2518 0.2332 0.2870 0.2642 0.2825 0.2119 0.1650 0.1516 0.1514 0.1428 0.1701 0.1579 0.1733 0.1617

1.90 1.86 1.54 1.61 2.21 1.75 1.94 2.11 2.19 1.99 3.82 3.66 3.73 3.62 3.44 3.31 3.62 3.50

228.9 216.0 123.1 139.8 363.2 180.6 246.5 316.6 350.2 264.1 1866.8 1643.3 1736.4 1590.0 1362.7 1218.6 1595.0 1438.4

1.69 2.56 2.76 2.30 1.79 2.45 1.79 1.79 1.62 2.37 1.59 1.80 1.77 1.93 1.71 1.91 1.59 1.77

1.80 2.41 2.58 2.21 1.87 2.32 1.87 1.87 1.76 2.26 1.74 1.87 1.85 1.96 1.82 1.94 1.75 1.85

1.54 2.71 2.47 − − 2.15 1.84 1.87 1.78 2.22 1.70 1.79 1.79 1.86 1.81 1.91 1.82 1.91

15.581 12.61b 12.621b 13.88 13.08 14.02b ″ 13.777b ″ 10.88b 8.44h 8.26k 8.24h 7.77k 8.94h 8.68k 9.14h 8.87k

c

11.54 16.52d 10.13d 9.61 19.45 13.18d ″ 16.92d ″ 18.69d 88.4j ″ 91.8j ″ 69.5j ″ 75.8j ″

4

The εR is according to eq 6 and εr to eq 17; Rmol is radius by eq 1, and Vmol = 3 π(2R mol)3. The last column contains molecular permittivity calculated from atomic components with scaling parameter kX = 0.5 (VH = 36.7, VC = 94.8, VN = 96.2, VO = 42.5, in au, hardness and polarizabilities in Table S1). The α and η, in au, are experimentally reported values (or theoretical by MP2/aug-cc-pVTZ, in italics). The (v.) stands for vertical and (a.) for adiabatic values. bRef 94. cRef 95. dRef 96. eRef 97. fRef 98. gRef 99. hRef 100. iRef 101. jRef 102. kRef 103. lRef 104. mRef 105. a

expressed via eq 18. However, the α itself has also been found to be dependent on density.106−108 Moreover, regardless of the state of matter, the results have to be corrected for dispersion (correlation and induced dipole−dipole interaction between neighboring atoms), which in the case of the gas phase is estimated to be 1.8%109 (undetermined for the condensed phase). Therefore, i.e., the dielectric constants of liquid or solid state noble gases89,90 significantly differ from gas-phase values89 (Table 1). The calculated dielectric constants of apolar species (without permanent dipole moment) very well agree with the experimentally measured values, after taking into account the conditions of the experiment by means of effective permittivity. Polar species, with dipole or high quadruple moments (higher moments are especially important in liquids and solids, due to small distances), might differ from reported experimental permittivity on account of their multiple moments and related additional temperature effects contributing to measured dielectric constant.

The scaling parameter, kX, is by least the consequence of cosharing of part of the electronic charge in a molecule between bonding atoms (similarly to an arbitrary division of population in the Mulliken scheme93). The valence electric charges extend throughout the whole molecule via molecular orbitals and thus experience inexplicable nuclear potential, which affects the resultant permittivity. If one isolate the atoms in a molecule, the effects related to interatomic charge transfer would suggest various oxidation states, followed by their unique polarizability and hardness and thus the radii and permittivity compared to neutral species. Therefore, the estimation of molecular properties from atomic components, especially involving strong interactions between them (such as covalent bonds), cannot be generalized and should be approached with skepticism. Nevertheless a satisfactory correlation might be observed between eq 6 and approximation by eq 20 for the presented group of molecules. Nearby Environment Effect on Dielectric Properties of Individual Systems. The molar volume includes specific thermodynamic conditions of pressure and temperature introduced through material density, affecting the permittivity F


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DISCUSSION In general, permittivity is a physical property dependent on temperature, pressure, and frequency. The sensitivity to these external influences can be attributed primarily to thermal particle rotations and vibrations causing permanent electric dipole reorientation. The temperature effects on nonpolar molecules are small and related mainly to variation in density and band gap width. Therefore, on account of the relations between dielectric constant and hardness (band gap, also through refractive index30−36) the temperature dependence might be coupled to the energy gap variation (in semiconductors it tends to decrease with raising temperature). Similarly the modulation of the interatomic distances, i.e., by applying high compressive (tensile) stress, affects the bandgap and results in dielectric constant sensitivity on the pressure or density as well. Still the permittivity is a tensor dependent on the frequency and external influences such as magnetic fields. Accordingly, it is separated into the static dielectric constant and the high frequency dielectric constant. Moreover, the measurable physical entity is resultant of static and induced effects; likewise the associated electric dipole moment can be split into permanent and induced parts. The permanent component of a dipole moment contributes to the dielectric constant through a molecule rotation (provided rotation time scales are comparable to the frequency of measurement). The induced part (almost instantaneous response) is due to migration of the electron density distribution around atoms, which exists up to very high frequencies. Since the permanent dipoles of molecules cannot rotate fast enough at high frequency fields, only the nonstatic part contributes at all frequencies. Local reorganization of the linked and free charges resulting in polarization phenomena of dielectric media can be expressed using the complex formulation of the dielectric permittivity, of which the real part describes storage of electromagnetic energy and the imaginary part reflects thermal conversion in relation to the frequency of the electromagnetic stimulation. The first part (energy storage) is proportional to the result of eq 6, if electronic polarizability is used, yet the temperature-related effects are also present through the band gap sensitivity. Following Maxwell’s theory of electromagnetic waves of material, it can be found that the square of the refractive index is equal to the relative permittivity measured at the same frequency. Depending on the frequency of electromagnetic fields, different types of charge oscillations in a material are induced: • inner or core electrons tightly bound to nuclei, • outer and/or valence electrons, free or conduction electrons, • bound ions in crystals, • free ions or multipoles. The storage of electromagnetic energy described through the dielectric constant defined in this work should therefore correspond to a critical frequency that induces distortions of inner and valence electronic shells. This suggests that the corresponding frequency should be found on the edge between the ultraviolet and optical ranges of the electromagnetic spectrum. Indeed, the comparison of results with the square of frequency-dependent refractive indices110 indicates the nearinfrared (NIR) frequency range, which directly points to valence electrons or more precisely to the highest occupied

orbitals (as implied also by the definitions in eq 6 and eq 10). However, it does not mean that other types of charge associations do not contribute at the same time. Supposing the electromagnetic field induced oscillations at some critical frequency become vanishingly small for uncharacteristic configurations, yet the lower the frequency the more of those modes are excited.

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CONCLUSIONS This simple model reveals important close relations between key electronic features of the systems at an individual atom or molecule level: the energy gap, electric dipole polarizability, and electric permittivity and their interdependencies on each other. This knowledge fundamentally supports the rational engineering of materials properties. The isotropic physical quantities have been described here; however, it is possible to extract anisotropic forms as well (simply by taking into account components of polarizability tensor and resulting set of radii, thus ellipsoidal shape of the respective volume, Figure 3). The

Figure 3. Ellipsoid of idealized volume of adenine, based on nonisotropic polarizability and molecular hardness; also the spherical volumes around the atoms according to their radii in the neutral state.

anisotropic permittivities simply result from molecule dimensions (scaling of eq 6 by the ratio of anisotropic radii to isotropic radius). Moreover, to some extent it is an additive property and might be composed of volume-weighted atomic (or groups) permittivities. Although there exist more advanced, abstract, and general methods to estimate dielectric permittivity of materials, the apprehensible concept presented in this work can successfully be used to provide an accurate description of physical and chemical phenomena. The original implementation of this idea might involve the limits of single-molecule design and interactions at the nanolevel (intra-, interatomic, -molecular). Besides it can find use in studies where the complexity and size of the system make it impossible to apply high level theories to solve the problem in question, like those described by coarsegrid models. Among other applications few examples might include improvement of solvation and solvent models,111 local electric field estimates, description of screening phenomena, or construction of molecular and single-molecule electronics and optoelectronic device studies. The difference between the G


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behavior of single species and a collection or between diluted and condensed phases is a highly important problem that is touched upon in this concept. The field-induced or -assisted reactivity has also been indicated.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b02626. Graphic comparison of dielectric constant estimates and experimental and numerical data for atoms (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +48 225540832. E-mail: [email protected]. ORCID

Paweł Szarek: 0000-0002-5706-0173 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author would like to dedicate this work in memory of Professor Józef Lipiński, outstanding scientist, scholar, and academic teacher for 40 years at Department of Chemistry, Wroclaw University of Technology. We lost the excellent educator of many generations of chemists and the pioneer of computational chemistry, our teacher, colleague, and friend. I greatly appreciate very helpful discussions with Professor Wojciech Grochala. This research has been funded by the National Science Center, grant NCN OPUS (UMO-2015/19/ B/ST4/02718). The use of resources of Wrocław Centre for Networking and Supercomputing (WCSS) and Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) grant No G49-17 and G49-23 is kindly acknowledged.

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