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Symbols, Units, and Constants

Equations are stated in terms of base units such as meters, kilograms, and seconds. Physical quantities in the text and tables are commonly described using scaled units such as nanometers, femtograms, and nanoseconds.

A ampere

AA amplitude; rate of change of amplitude; preexponential factor

A Hamaker constant (J)

a atto- (1018)\left(10^{-18}\right)

aa acceleration (m/s2)\left(\mathrm{m} / \mathrm{s}^{2}\right)

amu\mathrm{amu} atomic mass unit ( =1/12=1 / 12 mass of 12C1.661×1027 kg{ }^{12} \mathrm{C} \approx 1.661 \times 10^{-27} \mathrm{~kg} )

aJ attojoule (=1018 J)\left(=10^{-18} \mathrm{~J}\right)

B boron

Be beryllium

Br\mathrm{Br} bromine

bb exponential factor

c speed of light (3×108 m/s)\left(\sim 3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right); concentration (m3)\left(\mathrm{m}^{-3}\right)

C coulomb; carbon

Cl\mathrm{Cl} chlorine

CC London dispersion force coefficient (Jm6)\left(\mathrm{J} \cdot \mathrm{m}^{6}\right);
constant of integration;
compliance ( m/N)\mathrm{m} / \mathrm{N});

capacitance (F(\mathrm{F} )

Catten C_{\text {atten }} acoustic attenuation coefficient (s2/m)\left(\mathrm{s}^{2} / \mathrm{m}\right)

Cvol C_{\text {vol }} heat capacity per unit volume at constant volume (J/Km3)\left(\mathrm{J} / \mathrm{K} \cdot \mathrm{m}^{3}\right) Cn\mathscr{C}_{n} ratio of longitudinal contraction to transverse vibrational energy (m/J)(\mathrm{m} / \mathrm{J})

DD radiation dose ( rad\mathrm{rad} )

D0D_{0} binding energy of bond, with zero-point correction

DeD_{\mathrm{e}} binding energy of bond, without zero-point correction

Dsr D_{\text {sr }} coefficient of shearreflection drag

d distance (m); thickness (m)

dd^{\prime} dimensionless measure of interfacial stiffness

dad_{\mathrm{a}} interatomic distance along a line (m)

EE Young's modulus (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right); electric field strength (V/m)(\mathrm{V} / \mathrm{m})

EE_{\ell} rod elastic modulus (N)

E\mathscr{E} total energy (J)

eV\mathrm{eV} electron-volt (1.602×1019 J)\left(\approx 1.602 \times 10^{-19} \mathrm{~J}\right)

e elementary charge (1.602×1019C)\left(\approx 1.602 \times 10^{-19} \mathrm{C}\right)

F\mathrm{F} fluorine; farad

FF force (N)(\mathrm{N})

FF Helmholtz free energy (J)

f femto- (1015)\left(10^{-15}\right)

ff fractional quantity; frequency (cycles/s)

fTST f_{\text {TST }} frequency factor in transition state theory

fx(x)f_{x}(x) probability density function with respect to a variable xx

fxf_{x} spatial frequency along axis xx (cycles /m)/ \mathrm{m})

G giga- (109)\left(10^9\right)

GG shear modulus (N/m2)\left(\mathrm{N} / \mathrm{m}^2\right)

GG_{\ell} rod shear modulus (N)

G\mathscr{G} Gibbs free energy (J)

gcd greatest common denominator

HH hydrogen

He\mathrm{He} helium

Hz\mathrm{Hz} Hertz (= cycle/s)

H(fx)\left|\mathrm{H}\left(f_x\right)\right| amplitude spectral density of a potential along axis xx

hh henry

hh height (m)

\hbar Planck's constant

(1.055×1034 Js)\left(\approx 1.055 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)

ω\hbar \omega vibrational energy

quantum ( J\mathrm{J} )

I iodine

I moment of inertia (kgm2)\left(\mathrm{kg} \cdot \mathrm{m}^2\right)

ii an integer (often an index);
(1)1/2(-1)^{1 / 2}

J\mathrm{J} joule

JJ exchange integral (J)(\mathrm{J})

jj an integer (often an index)

K\mathrm{K} degree kelvin

KdK_{\mathrm{d}} dissociation constant (m3)\left(\mathrm{m}^{-3}\right)

kg kilogram

kk Boltzmann's constant (1.381×1023 J/K)\left(\approx 1.381 \times 10^{-23} \mathrm{~J} / \mathrm{K}\right);
optical extinction coefficient

k12k_{12} rate of transitions from state 1 to state 2, given occupancy of state 1( s1)1\left(\mathrm{~s}^{-1}\right)

kak_{\mathrm{a}} \quad stiffness per unit area of interface (N/m3)\left(\mathrm{N} / \mathrm{m}^3\right)

kbk_{\mathrm{b}} bending stiffness of a beam (Jm/rad2)\left(\mathrm{J} \cdot \mathrm{m} / \mathrm{rad}^2\right)

kcubic k_{\text {cubic }} MM2 cubic bond stretching constant (m1\left(\mathrm{m}^{-1}\right. )

kerr k_{\text {err }} erroneous reaction rate (s1)\left(\mathrm{s}^{-1}\right)

kisc k_{\text {isc }} intersystem crossing rate (s1)\left(\mathrm{s}^{-1}\right)

kk_{\ell} transverse stiffness per unit length (N/m2)\left(\mathrm{N} / \mathrm{m}^2\right)

kreact k_{\text {react }} correct reaction rate (s1)\left(\mathrm{s}^{-1}\right)

ksk_{\mathrm{s}} mechanical stiffness (N/m2)\left(\mathrm{N} / \mathrm{m}^2\right)

ksextic k_{\text {sextic }} MM2 sextic bond angle-bending constant (rad4)\left(\mathrm{rad}^{-4}\right)

ksθk_{\mathrm{s} \theta} stretch-bend parameter (N/rad)

ksk_{\mathrm{s} \perp} mechanical stiffness perpendicular to a bond or axis (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right)

kTk T characteristic thermal energy ( JJ )

kt,bk_{\mathrm{t}, \mathrm{b}} transverse-displacement stiffness of a bending beam (N/m)(\mathrm{N} / \mathrm{m})

knk_{n} stiffness of vibrational mode n( N/m=J/m2)n\left(\mathrm{~N} / \mathrm{m}=\mathrm{J} / \mathrm{m}^{2}\right)

kθk_{\theta} angular spring constant (J/rad2)\left(\mathrm{J} / \mathrm{rad}^{2}\right)

k wave vector (m1)\left(\mathrm{m}^{-1}\right); magnitude of wave vector (m1)\left(\mathrm{m}^{-1}\right)

kD\boldsymbol{k}_{\mathrm{D}} Debye radius (in reciprocal space) (m1)\left(\mathrm{m}^{-1}\right)

L\mathrm{L} ligand

Li lithium

LP lone pair

LL length in scaling relationships (m); inductance (h)(\mathrm{h})

lcm\mathrm{lcm} least common multiple

\ell length (m)

M molar (103 mole/m3)\left(10^{3} \mathrm{~mole} / \mathrm{m}^{3}\right); metal atom

MIPS million instructions per second

MM modulus of elasticity (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right)

MsM_{\mathrm{s}} Mach number (=v/vs)\left(=v / v_{\mathrm{s}}\right)

m\mathrm{m} meter; milli- (103)\left(10^{-3}\right)

maJ milli-attojoule (1021 J)\left(10^{-21} \mathrm{~J}\right)

mole gram mole (=6.022×1023)\left(=6.022 \times 10^{23}\right)

mm mass (kg)(\mathrm{kg}); integer

N\mathrm{N} newton (kgm/s2\left(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\right. ); nitrogen

NN integer

n nano- (109)\left(10^{-9}\right)

nm\mathrm{nm} nanometer (109 m)\left(10^{-9} \mathrm{~m}\right)

nn integer; refractive index

nan_{\mathrm{a}} areal number density (m2)\left(\mathrm{m}^{-2}\right)

nvn_{\mathrm{v}} volumetric number density (m2)\left(\mathrm{m}^{-2}\right)

OO oxygen

PP phosphorus

Pa\mathrm{Pa} pascal (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right)

PP probability; power (W)

Perr P_{\text {err }} probability of error

P(x)P(x) probability of condition xx

p pico- (1012)\left(10^{-12}\right) pKapH\mathrm{p} K_{\mathrm{a}} \quad \mathrm{pH} at which 50%50 \% of the molecules of an acidic species are dissociated

p pressure (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right); magnitude of momentum (kgm/s)(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s})

pip_{i} momentum coordinate ii (kgm/s)(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s})

P momentum vector (kgm/s)(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s})

QQ Coulomb integral ( J\mathrm{J} )

qq electrical charge (C); generalized coordinate

qq partition function

RR alignment band velocity ratio;
Reynolds number;
concentration ratio;
resistance (Ω)(\Omega)

Rtemp R_{\text {temp }} compression temperature ratio

Rnn\mathscr{R}_{n} \quad nth root of Eqs. (5.45) and (5.46) describing beam vibrations;
rad radians;
radiation dose (102 J/kg)\left(10^{-2} \mathrm{~J} / \mathrm{kg}\right)

rr distance or radius (m);
interest rate (yr1)\left(\mathrm{yr}^{-1}\right)

r0r_{0} reference distance or radius

rir_{i} position coordinate i( m)i(\mathrm{~m})

rvdw0 r_{\text {vdw0 }} MM2 van der Waals radius

r coordinate vector in configuration space (m)

SS sulfur

Si silicon

SN2S_{N} 2 nucleophilic substitution by direct displacement

SS area (in real space, m2\mathrm{m}^{2}; in nn-dimensional configuration space, mn1)\left.\mathrm{m}^{n-1}\right)

S\mathscr{S} entropy (J/K)(\mathrm{J} / \mathrm{K})

s second

ss separation (m)

T\mathrm{T} tesla

TT temperature (K);
transmittance; torque (Nm)(\mathrm{N} \cdot \mathrm{m})

TT/TDT^{\prime} \quad T / T_{\mathrm{D}}

T300300 KT_{300} 300 \mathrm{~K}

TDT_{\mathrm{D}} Debye temperature (K)

Ttrans T_{\text {trans }} phonon transmission coefficient

T\mathscr{T} kinetic energy ( J\mathrm{J} )

tt time (s) tact t_{\text {act }} actuation time

ttrans t_{\text {trans }} transformation time

V\mathrm{V} volt

VV volume (in real space, m3\mathrm{m}^{3};
in nn-dimensional configuration space, mn\mathrm{m}^{n} )

\checkmark potential energy (often a function of position coordinates) ( J\mathrm{J} )

V\mathcal{V}^{\prime} potential energy of a well, corrected for zero point vv speed (m/s)(\mathrm{m} / \mathrm{s})

V\mathcal{V}^{\dagger} potential energy including zz-axis elastic constraint ( J\mathrm{J} )

ΔV\Delta \mathcal{V}^{\ddagger} barrier height (transition-state energy minus well energy) (J)

ΔV\Delta \mathscr{V}^{\prime \ddagger} barrier height based on corrected well energy ( J)\mathrm{J})

Vω\mathscr{V}_{\omega} potential energy of bond torsion

VθV_{\theta} potential energy of bond angle-bending speed (m/s)(\mathrm{m} / \mathrm{s})

vsv_{\mathrm{s}} \quad speed of sound (m/s)(\mathrm{m} / \mathrm{s})

WW work (J)

ww width (m)

xx spatial coordinate (m)(\mathrm{m})

yr year (3.154×107 s)\left(\approx 3.154 \times 10^{7} \mathrm{~s}\right)

y(x)y(x) displacement as a function of xx

yy spatial coordinate (m)(\mathrm{m}); transverse displacement (m); general-purpose variable

zz spatial coordinate (m); general purpose variable

α\alpha accommodation coefficient; selective transport coefficient

β\beta Morse function scaling parameter (m1)\left(\mathrm{m}^{-1}\right);
volume coefficient of thermal expansion (K1)\left(\mathbf{K}^{-1}\right)

Γ\Gamma^{*} Wigner tunneling correction factor

γ\gamma Shear stress (N/m2)\left(\mathrm{N} / \mathrm{m}^{2}\right)

γG\gamma_{\mathrm{G}} Grüneisen number

γ\gamma_{\ell} tension in rod (N)

δ\delta positional tolerance (m); minimum imaging separation (m)

δsurf  distance between structural-  and excluded-volume  boundaries (m)x/y derivative of x with respect  to y, holding other  variables constant ε zero-frequency dielectric  constant; phonon energy  density (J/m3) surface contour tolerance (m) ε0 electrical permittivity of free  space (8.854×1012 F/m)εvdw  MM2 van der Waals energy η viscosity (Ns/m2)ηphonon  phonon viscosity (Ns/m2)θ angle (rad); bond angle (rad) θ0 reference angle (rad) θsym  minimum rotational symmetry  angle (rad) κn spatial frequency factor in  beam bending equations λ wavelength (m) μ reduced mass (kg) micro- (106); micron (106 m)\begin{aligned} & \delta_{\text {surf }} \text { distance between structural- } \\ & \text { and excluded-volume } \\ & \text { boundaries }(\mathrm{m}) \\ & \partial x / \partial y \text { derivative of } x \text { with respect } \\ & \text { to } y \text {, holding other } \\ & \text { variables constant } \\ & \varepsilon \text { zero-frequency dielectric } \\ & \text { constant; phonon energy } \\ & \text { density }\left(\mathrm{J} / \mathrm{m}^{3}\right) \text {; } \\ & \text { surface contour tolerance (m) } \\ & \varepsilon_{0} \text { electrical permittivity of free } \\ & \text { space }\left(8.854 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right) \\ & \varepsilon_{\text {vdw }} \text { MM2 van der Waals energy } \\ & \eta \text { viscosity }\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right) \\ & \eta_{\text {phonon }} \text { phonon viscosity }\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right) \\ & \theta \text { angle (rad); bond angle (rad) } \\ & \theta_{0} \text { reference angle (rad) } \\ & \theta_{\text {sym }} \text { minimum rotational symmetry } \\ & \text { angle (rad) } \\ & \kappa_{n} \text { spatial frequency factor in } \\ & \text { beam bending equations } \\ & \lambda \text { wavelength (m) } \\ & \mu \text { reduced mass }(\mathrm{kg}) \text {; } \\ & \text { micro- }\left(10^{-6}\right) \text {; micron }\left(10^{-6} \mathrm{~m}\right) \end{aligned}

vv Poisson's ratio

ρ\rho mass density (kg/m3)\left(\mathrm{kg} / \mathrm{m}^{3}\right)

ρ\rho_{\ell} linear mass density (kg/m)(\mathrm{kg} / \mathrm{m})

σ\sigma standard deviation in position (m); scattering cross section (m2)\left(\mathrm{m}^{2}\right)

ϕ\phi phase angle (rad); spherical coordinate angle (rad)

ϕw\phi_{\mathrm{w}} work function (eV)

ψ\psi spherical coordinate angle (rad)

ψ(x)\psi(x) wave function (commonly complex-valued)

ψ(x)\psi^{*}(x) complex conjugate of ψ(x)\psi(x)

Ω\Omega ohm

ω\omega angular frequency (rad/s); torsion angle (rad)

ω0\omega_{0} reference torsion angle (rad)

ωrc \omega_{\text {rc }} imaginary frequency of motion along reaction coordinate (s1\left(\mathrm{s}^{-1}\right. )

\textdollar U.S. dollar, 1990