Units (v.1.6.0) Andrea Dal Corso (SISSA - Trieste) 1 Contents 1 Introduction 4 2 People 5 3 Overview 6 4 Time 8 5 Length 9 6 Mass 10 7 Mass density 11 8 Frequency 12 9 Speed 13 10 Acceleration 14 11 Momentum 15 12 Angular momentum 16 13 Force 17 14 Energy 18 15 Power 19 16 Pressure 20 17 Temperature 21 18 Current 22 19 Charge 24 20 Charge density 26 21 Current density 27 22 Electric field 28 23 Electric potential 29 24 Capacitance 30 25 Vacuum electric permittivity 31 2 26 Electric dipole moment 32 27 Polarization 33 28 Electric displacement 34 29 Resistance 36 30 Magnetic flux density 37 31 Vector potential 39 32 Magnetic field flux 40 33 Inductance 41 34 Magnetic dipole moment 42 35 Magnetization 43 36 Vacuum magnetic permeability 44 37 Magnetic field strength 45 38 Microscopic Maxwell’s equations 48 39 Macroscopic Maxwell’s equations 49 40 The Schrödinger equation 50 41 Appendix A: Rydberg atomic units 51 42 Appendix B: Gaussian atomic units 55 43 Appendix C: Magnetization Intensity 57 44 Appendix D: Conversion factors tables 57 45 Bibliography 65 3 1 Introduction These notes discuss the atomic units (a.u.) used in electronic structure codes. They are updated with the recent (year 2019) changes to the international system (SI). The conversion factors written here should be those implemented in the QUANTUM ESPRESSO and thermo_pw codes. These notes are part of the thermo_pw package. The complete package is available at https://github.com/dalcorso/thermo_pw. 4 2 People These notes have been written by Andrea Dal Corso (SISSA - Trieste). Disclaimer: I am not an expert of units. These notes reflect what I think about units. If you think that some formula is wrong, that I misunderstood some- thing, or that something can be calculated more simply, please let me know, I would like to learn more. You can contact me directly: [email protected]. 5 3 Overview Electronic structure codes use atomic units (a.u.). In these notes we explain how to obtain the conversion factors from a.u. to the international system (SI) units and viceversa. The most relevant physical formulas are written both in SI units and in a.u.. Moreover, since several books and old literature still use the c.g.s. (centimeter-gram-second) system we discuss also how to convert from a.u. to c.g.s.-Gaussian units and viceversa. These notes are organized in Sections, one for each physical quantity. For each quantity we give its definition in the SI system and then we derive the conversion factor from the a.u. to the SI unit, the conversion factor from the c.g.s-Gaussian unit to the SI unit, and the conversion factor from the a.u. to the c.g.s-Gaussian unit. The text in each Section depends on the defini- tions given in previous Sections. Only in a few cases it depends on definitions given in following Sections and in these cases we mention explicitly where to find the required definition. Important physical formulas are introduced when we have given sufficient information to convert the formula in different sys- tems. Microscopic and macroscopic Maxwell’s equations are also summarized in separate Sections. Each system has a different color. In black the SI, in blue the a.u., in orange the c.g.s.-Gaussian system, and in green the conversion factors from a.u. to the c.g.s.-Gaussian units. Comments of interest not belonging to any sys- tem in particular are given in red. We indicate the numerical value of a given quantity in the SI with a tilde. The units in a.u. are indicated with a bar, while the units in the c.g.s.-Gaussian system are indicated with a bar and the subscript cgs. When the c.g.s.-Gaussian unit has an accepted name we use it interchangeably with the generic name. Only Hartree a.u. are described in the main text since these are the most common microscopic units. QUANTUM ESPRESSO uses Rydberg a.u. that can be easily derived from the Hartree a.u.. We describe Rydberg a.u. in Appendix A (in violet). In the ab − initio literature modified a.u. have been introduced in which the electromagnetic equations look like those of the c.g.s-Gaussian system. This requires some modifications to the definitions given in the main text and we discuss these units in Appendix B (in steelblue). It is useful to recall a few preliminary facts needed in the rest of these notes. A few experimental quantities have fixed values in the SI. Among these: The Planck constant h = 6:62607015 × 10−34 J · s; (1) the speed of light c = 2:99792458 × 108 m=s; (2) and the electron charge e = 1:602176634 × 10−19 C: (3) Two experimental quantities are known with high accuracy: The Rydberg constant, measured spectroscopically from the frequencies of 6 Hydrogen and Deuterium absorption and emission lines: 7 R1 = 1:0973731568160 × 10 1=m (4) is known with a relative error of 1:9 × 10−12 and the fine structure constant, measured from the anomaly of the electron magnetic moment, α = 7:2973525693 × 10−3; (5) is known with a relative error of 1:6 × 10−10. The Rydberg constant can be written in terms of the fine structure constant α and the electron mass me or in terms of α and of the Bohr radius aB: 2 α mec α R1 = = : (6) 2h 4πaB Using these equations we can calculate the electron mass as: 2hR m = 1 = 9:1093837015 × 10−31 kg (7) e α2c and aB as: α −11 aB = = 5:29177210903 × 10 m: (8) 4πR1 α is a pure number equal to: e2 µ ce2 α = = 0 ; (9) 4π0hc¯ 2h where h¯ = h=2π, 0 the vacuum electric permittivity and µ0 is the vacuum magnetic permeability. Hence the Bohr radius can be rewritten as: 2 h¯ h¯ 4π0 aB = = 2 : (10) αmec mee Eq. 9 allows the calculation of µ0 as: 2hα N µ = = 1:25663706212 × 10−6 : (11) 0 e2c A2 0 is obtained from the relation: 2 1 −12 C 0 = 2 = 8:8541878128 × 10 2 : (12) µ0c N · m It is also useful to define the hartree energy: 2 e −18 Eh = = 2hcR1 = 4:3597447222072 × 10 J; (13) 4π0aB and the Bohr magneton: he¯ −24 µB = = 9:2740100783 × 10 J=T: (14) 2me 7 4 Time In the SI, the unit of time is the second (symbol s), defined requiring that the frequency of a particular line of the 133Cs atom is exactly 9192631770 s−1. In a.u. the unit of time (symbol ¯t) is defined requiring that the numerical value of h¯ is 1. The conversion factor with the SI unit is obtained recalling that in the SI the Planck constant is h¯ = h¯~ J · s = h¯~ kg · m2=s. Therefore in a.u. we have: ¯m ·¯l2 h¯ = ; (15) ¯t ¯ ¯ where l is the unit of length and ¯m the unit of mass. As we discuss below l = aB and ¯m= me, therefore m a2 h¯4π a h¯ 1:0545718176462 × 10−34 J · s ¯ e B 0 B −17 t = = 2 = = −18 = 2:4188843265857×10 s h¯ e Eh 4:3597447222072 × 10 J (16) In the c.g.s. system the unit of time is the second (symbol s), defined as in the SI. We have ¯tcgs = s. The conversion factor between the a.u. and the c.g.s. unit is: ¯t = 2:4188843265857× 10−17 s. 8 5 Length In the SI, the unit of length is the metre (symbol m) defined as the length of the path traveled by light during the time interval of 1=299792458 s. In a.u. the unit of the length (symbol ¯l) is defined requiring that the Bohr ra- ¯ ¯ −11 dius aB = l. The conversion factor with the SI unit is l = 5:29177210903×10 m. In the c.g.s. system the unit of length is the centimetre (symbol cm) defined −2 ¯ −2 as cm = 1:0 × 10 m. We have lcgs = 10 m. The conversion factor between the a.u. and the c.g.s. unit is: ¯l = 5:29177210903× 10−9 cm. A common unit of length is the angstrom (symbol Å). We have Å= 10−10 m. 9 6 Mass In the SI the unit of mass is the kilogram (symbol kg) defined requiring that the Planck constant h has the value 6:62607015 × 10−34 kg · m2=s. In a.u. the unit of mass (symbol ¯m) is defined requiring that the mass of the electron me = ¯m. The conversion factor with the SI unit is ¯m= 9:1093837015 × 10−31 kg. In the c.g.s. system the unit of mass is the gram (symbol g) defined as g = 1:0 × 10−3 kg. The conversion factor between the a.u. and the c.g.s. unit is ¯m= 9:1093837015× 10−28 g. Atomic weights are usually expressed in atomic mass units (a:m:u: = 1:66053906660 × 10−27 kg). This quantity can be converted in a.u. as: 1:66053906660 × 10−27 kg a:m:u: = ¯m= 1:8228884862 × 103 ¯m (17) 9:1093837015 × 10−31 kg 10 7 Mass density In the SI the unit of mass density is derived from its definition: dm ρ = ; (18) m dV and it is kg=m3. In a.u. the unit of mass density (symbol ρ¯m) is derived from its definition: ¯m ρ¯m = ¯l3 .