Relativistic Corrections to The Voltage of Lead-Acid Batteries: Matthew Mason, University of Chicago: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KVEpQWJzdHJhY3RGJy8lJXNpemVHUSMxOEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRj1GLy8lK2V4ZWN1dGFibGVHRjcvRjxRJ25vcm1hbEYn This worksheet will begin with a quick introduction to implementing relativity into quantum mechanics. The worksheet will then provide a quick summary on how to practically perform relativistic quantum calculations. The worksheet will then use the quantum chemistry package to calculate the energy level of the 6s orbital of lead using the nonrelativistic Hartree Fock theory. This orbital energy is then compared to the relativistic results of the 6s orbital energy. The differences between these orbital energies are used to explain the significant role special relativity plays in the voltage of Lead-acid batteries.

Dimitri Mendeleev created the periodic table in 1869 to arrange the elements into periods of similar chemical and physical characteristics. The periodic table works well for the lighter elements but starts to fail in its predictions as the elements get heavier. Some examples are that mercury is a liquid at room temperature, lead significantly outperforms tin in metal-acid batteries, and the color of gold metal. One hint as to why the predictions of the periodic table start to fail for heavier atoms are that quantum mechanics predicts the speed of electrons for heavy atoms to be close to the speed of light. According to the virial theorem a stable system of discrete particles bound by potential forces must satisfy the following equation where E is total energy, T is kinetic energy, and U is potential energy [1]. 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 This gives us a relationship between the velocity of the electron 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 and the energy of the electron in a 1s orbital in atomic units. 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 The relationship shows the velocity is proportional to the nuclear charge. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZwcm9wO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIlpGJ0YvRjIvJStleGVjdXRhYmxlR0Y9Rjk= The speed of light in atomic units is 137.036. This proportionality relation shows that quantum mechanics predicts that core electrons of heavy atoms approach speeds close to the speed of light. The relationship provided between nuclear charge and electron velocity cannot always hold, since it does not put a limit on the speed of the electrons. According to Einstein\342\200\231s special theory of relativity, the fastest anything can move is the speed of light, so any theory that allows for speeds greater than the speed of light are not relativistic. This quick calculation demonstrates that the Schrodinger equation is not a relativistic theory and that relativistic effects are likely significant for heavy atoms. For a theory to be relativistic, it must be invariant under a Lorentz transformation. The Lorentz transformation is given as follows. 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 A new theory is needed that merges the properties of quantum mechanics and special relativity. This new theory would ideally be able to explain the peculiarities of heavy atoms and allow researchers to make predictions of the behavior and properties of systems involving heavy atoms. Fortunately, Paul Dirac derived such an equation. The next section describes the Dirac equation which is a relativistic quantum mechanical equation that describes electrons.

In 1928 Paul Dirac derived the Dirac Equation, which is a relativistic equation that describes free electrons. The Dirac Equation is as follows. 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 In the Dirac equation c\316\261 is the relativistic velocity operator, p is the momentum, m is the relativistic mass, and \317\203 are the Pauli matrices. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYoLUZKNiVRInhGJ0ZNRlAtRiw2LVEiLEYnRi9GMi9GNkZPRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZKNiVRInlGJ0ZNRlBGVy1GSjYlUSJ6RidGTUZQRi8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmRvLUkobWFjdGlvbkdGJDYkLUkobWZlbmNlZEdGJDYoLUYjNiYtRko2I1EhRictSSdtdGFibGVHRiQ2Ny1JJG10ckdGJDYoLUkkbXRkR0YkNiktRiM2JC1JI21uR0YkNiRGX29GL0YvLyUrZXhlY3V0YWJsZUdGNC8lKXJvd2FsaWduR0ZgcC8lLGNvbHVtbmFsaWduR0ZgcC8lK2dyb3VwYWxpZ25HRmBwLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRmlxLUZocDYpLUYjNiQtRkc2JS1GSjYlUSgmc2lnbWE7RicvRk5GNEYvRlJGXW9GL0ZfcUZhcUZjcUZlcUZncUZqcUZfcUZhcUZjcUZlcS1GZXA2KEZcckZncEZfcUZhcUZjcUZlcUZfcS8lJmFsaWduR1ElYXhpc0YnL0ZicVEpYmFzZWxpbmVGJy9GZHFRJ2NlbnRlckYnL0ZmcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRk8vJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zlcy8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZgdC8lJmZyYW1lR0ZgdC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRjQvJS1lcXVhbGNvbHVtbnNHRjQvJS1kaXNwbGF5c3R5bGVHRjQvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGXXRGXnBGL0YvL0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGY3UvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnRitGK0ZfcUYv 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 When comparing energy calculations using the Dirac equation to energy calculations with the Schr\303\266dinger equation, the zero point of the energy scale needs to be shifted so that both calculations match. This is due to the rest energy calculated by the Schrodinger equation being 0 and the rest energy calculated by the Dirac equation being LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYmLUkjbWlHRiQ2JlEjbWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiVRIjJGJ0Y1L0Y5USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSJeRidGNUY/. The energy scale of the Dirac equation is typically shifted to match the Schrodinger equation by replacing \316\262 with \316\262'. 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 Solutions to the Dirac equation have four components and take the following form. 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 Each of these components has an important physical meaning. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjFGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw== is an electron with a positive spin, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjJGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw== is an electron with a negative spin, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjNGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw== is a positron with a positive spin, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjRGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw== is a positron with a negative spin. The positronic states have energies that lie below 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. Since the positronic and electronic states are physically significantly different, they are usually grouped separately from each other. The positronic states are grouped into what are called the small components (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1GLzYmUSJTRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y/RjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw==), meanwhile, the electronic states are grouped together as the large components (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1GLzYmUSJMRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y/RjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw==).

The Dirac equation discussed in the last section is for free electrons. This equation is important for physics, but not relevant for the purposes of describing the chemistry and physical properties of atoms. Fortunately, the Dirac equation can be elaborated on to incorporate external electric and magnetic potentials. The Dirac equation that describes electrons in the presence of external electric potentials takes the following form, where V is the external electric potential. 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 Since \316\261 and \316\262' are block diagonal matrices, the Dirac Equation can be factored into two coupled equations. 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 These equations can be solved to get the following relationship. 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 With this relationship we can write the Dirac Equation for the Large components as. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYmLUYjNistSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYnLUY0NiRRIjJGJ0Y3LUkjbW9HRiQ2LVEifkYnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1JI21pR0YkNiVRIm1GJy8lJ2l0YWxpY0dRJXRydWVGJy9GOFEnaXRhbGljRidGZW5GaG4vJS5saW5ldGhpY2tuZXNzR0Y2LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5vLyUpYmV2ZWxsZWRHRkVGPy1GLDYkLUYjNictRlg2JVEnJiM5NjM7RicvRmZuRkVGNy1GQDYtUScmc2RvdDtGJ0Y3RkNGRkZIRkpGTEZORlBGUkZVLUZYNiVRInBGJ0ZlbkZobi8lK2V4ZWN1dGFibGVHRkVGN0Y3LUZYNiVRIktGJ0ZlbkZobkZjby1GQDYtUSIrRidGN0ZDRkZGSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmpwLUYsNiQtRiM2Jy1GWDYlUSJWRidGZW5GaG4tRkA2LVEqJnVtaW51czA7RidGN0ZDRkZGSEZKRkxGTkZQRmlwRltxLUZYNiVRIkVGJ0ZlbkZobkZhcEY3RjdGYXBGN0Y3LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRj8tSSVtc3ViR0YkNiUtRlg2JVEmJnBzaTtGJ0Zqb0Y3LUYjNiQtRlg2JVEiTEYnRmVuRmhuRjcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GQDYtUSI9RidGN0ZDRkZGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmFzLUY0NiRGXHNGN0ZhcEY3 At this point it is important to check if the Dirac equation has the properties we expect, specifically the Dirac equation should become the Schrodinger equation in the limit where the speed of light is set to infinity. Setting the speed of light to be infinity makes K=1, and 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 transforming the Dirac equation into the following form. 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 This is the Schrodinger equation, so the Dirac equation matches the Schrodinger equation in the appropriate limit. The Dirac equation in the presence of Electric potentials works great when there are no magnetic fields present, but in complex systems such as batteries where there are many moving charges, incorporating magnetic fields is essential. Accounting for the magnetic fields is quite simple. The procedure is to just replace the momentum operator with the generalized momentum operator. 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 A is the magnetic vector potential defined by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmbmFibGE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjExMTExMTFlbUYnLUY2Ni1RKCZ0aW1lcztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GWi1GLDYlUSJBRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5 Incorporating magnetic potentials turns the Dirac equation into 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 This relativistic equation captures the quantum properties of single electrons in the presence of Magnetic and Electric fields. For the needs of chemistry, describing single electrons is not powerful enough. To apply relativistic corrections to quantum chemistry, a many-electron Dirac equation is needed.

Like the non-relativistic case, the many-electron equation for relativistic quantum mechanics quickly becomes incredibly complex as more electrons are added to the system. In the relativistic case, the potential part of the many-electron equation does not have a closed form [1]. A common approximation that makes the many-electron Dirac equation more manageable, is to Taylor expand the potential in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiZRImNGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y0RjJGNQ== up to terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1zdXBHRiQ2Ji1JI21pR0YkNiZRImNGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJy1GLzYlUSIyRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRIl5GJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR0Y0RjJGNQ== . Under this approximation, the electron-electron repulsion takes the following form and is known as the Coulomb-Breit potential. 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 The Dirac Hamiltonian in addition to the Coulomb-Breit potential can be used to formulate many operators that correspond to certain relativistic corrections. There are an infinite number of corrections, but only 13 that are of the order of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1zdXBHRiQ2Ji1JI21pR0YkNiZRImNGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJy1GLzYlUSIyRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRIl5GJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR0Y0RjJGNQ== or less. One correction is the mass-velocity correction. This correction incorporates the electron\342\200\231s mass being dependent on its velocity and is given as follows. 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 There are three corrections that arise from single electrons interacting with external fields. These corrections are the Zeeman, spin-orbit, and Darwin corrections and are given as follows. In these corrections, F is the electric field, B is the magnetic field, s is the electron spin, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImVGJ0YyRjUvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYsNiUtRi82JVElJm11O0YnL0YzUSZmYWxzZUYnRkAtRiM2JS1GLzYlUSJCRidGMkY1Rj1GQEZCRj1GQA== is a corrective factor to the electron spin. 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 There are five corrections that arise from two electron interactions. These corrections are the spin-orbit, spin-other-orbit, spin-spin, orbit-orbit, and the Darwin correction. The spin-orbit correction accounts for an electrons spin interacting with the magnetic field produced by its movement. The spin-other-orbit correction accounts for the electron spin interacting with the magnetic fields produced by other electrons. The spin-spin and orbit-orbit account for other magnetic interactions, whereas the Darwin term accounts for the electric repulsion of the electrons. These corrections are given as the following. 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There are also 4 corrections due to the electrons interacting with the nuclei. These corrections are the spin-orbit, paramagnetic spin-orbit, spin-spin, and Darwin corrections. I in these equations is the nuclear spin LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYlUSJBRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiUtRiw2JVEiTkYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is the relativistic correction to the nuclear spin. The electron-nuclei relativistic corrections are given as follows. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI25lRidGMkY1RjJGNS1GIzYlLUYvNiVRJ0RhcndpbkYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUkjbW9HRiQ2LVEiPUYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUi8lKXN0cmV0Y2h5R0ZSLyUqc3ltbWV0cmljR0ZSLyUobGFyZ2VvcEdGUi8lLm1vdmFibGVsaW1pdHNHRlIvJSdhY2NlbnRHRlIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zbby1JJm1mcmFjR0YkNigtRi82JVElJnBpO0YnL0YzRlJGTi1GIzYoLUkjbW5HRiQ2JFEiMkYnRk4tRks2LVEifkYnRk5GUEZTRlVGV0ZZRmVuRmduL0ZqblEmMC4wZW1GJy9GXW9GX3AtSSVtc3VwR0YkNiUtRi82JVEibUYnRjJGNS1GIzYlRmdvRjJGNUZCLUZicDYlLUYvNiVRImNGJ0YyRjVGZ3BGQkYyRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNxLyUpYmV2ZWxsZWRHRlItSSttdW5kZXJvdmVyR0YkNictRks2LVEmJlN1bTtGJ0ZOL0ZRUSZ1bnNldEYnL0ZURl9yL0ZWRjQvRlhGX3IvRlpGNC9GZm5GNC9GaG5GX3JGXnAvRl1vUSwwLjE2NjY2NjdlbUYnLUYjNiYtRi82JVEiaUYnRjJGNUZKLUZobzYkRmBxRk5GTi1GIzYkLUklbXN1YkdGJDYlLUYvNiVRIk5GJ0YyRjUtRiM2JC1GLzYlUSVlbGVjRidGMkY1Rk5GRUZORmduLyUsYWNjZW50dW5kZXJHRlItRmlxNidGW3ItRiM2Ji1GLzYlUSJBRidGMkY1RkpGXXNGTi1GIzYkLUZiczYlRmRzLUYjNiQtRi82JVEnbnVjbGVpRidGMkY1Rk5GRUZORmduRlx0LUZiczYlLUYvNiVRIlpGJ0YyRjUtRiM2JUZidEYyRjVGRS1GLzYlUScmIzk0ODtGJ0Zkb0ZOLUkobWZlbmNlZEdGJDYkLUYjNiUtRmJzNiUtRi82JVEickYnRjJGNS1GIzYlLUYvNiVRI2lBRidGMkY1RjJGNUZFLyUrZXhlY3V0YWJsZUdGUkZORk5GZ3ZGTg== There are more corrections, but they are proportional to higher powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiZRImNGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y0RjJGNQ==than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1zdXBHRiQ2Ji1JI21pR0YkNiZRImNGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJy1GLzYlUSIyRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRIl5GJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR0Y0RjJGNQ==. Having all these corrections are great, but exactly implementing them into calculations of realistic systems is not always practical. The next sections will discuss how to practically perform relativistic quantum mechanical calculations of many electron systems.

There are many methods to practically perform relativistic quantum mechanical calculations with varying approximations, degrees of complexity, and levels of accuracy. One approach is to treat relativistic effects as perturbations on non-relativistic quantum mechanics. This method is relatively low on accuracy, but not as computationally intensive as methods that will be described soon. Perturbative relativity is done by treating the corrections described in the last section as perturbations on the Hamiltonian of the Schrodinger equation. When conducting this method, it is important to note that some of these corrections are divergent, so they cannot be implemented past 1st order perturbation theory [1].

Other techniques that implement relativistic behavior into quantum mechanics tend to solve the Dirac equation under certain approximations. One example is four component calculations. For four component calculations it is assumed that the total wavefunction can be written as a single slater determinant of spinors. Spinors are four component one electron functions that serve the same purpose as spin-orbitals in non-relativistic theories [1]. The equation that is used for four-component calculations is the time-independent Dirac-Fock equation which is given as. 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 Solving the Dirac-Fock equation involves navigating some of its intricacies. One of these is how it limits basis sets. The small components and the large components each use their own basis sets. The large components basis sets have the same requirements as in the non-relativistic case [1]. The small component basis set must be related to the large component basis set, in a specific way that allows the solution to smoothly transform into the non-relativistic solution as the speed of light increases [1]. This condition is called the kinetic balance condition and is given as follows. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYmLUkjbWlHRiQ2JlEmJmNoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLzYmUSZzbWFsbEYnL0YzUSV0cnVlRidGNS9GOFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEiXkYnRjVGNw==is the small component basis set and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYmLUkjbWlHRiQ2JlEmJmNoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLzYmUSZsYXJnZUYnL0YzUSV0cnVlRidGNS9GOFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEiXkYnRjVGNw== is the large component basis set. 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 Another intricacy of the Dirac-Fock equation is how it treats the charge distribution of nuclei. If nuclei are treated as point charges, the non-relativistic calculation will have the orbitals have a cusp at the nuclei. This is manageable, but the relativistic case provides a singularity at the nuclei. Singularities are not manageable, so a different assumption for the charge distribution of nuclei must be used. It is common to have the nuclei have a finite radius or be a three-dimensional gaussian. Four-Component calculations give the most accurate relativistic corrections but are computationally expensive.

Four component calculations can be computationally expensive, so it is common to perform a two-component calculation instead. The two-component method decouples the large and small components and then finds the solution to just the large component. The decoupling is done through an energy independent unitary transformation of the Dirac operator. The Dirac operator and its transformed version take the form of. 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 The transformed wavefunction is then. 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 What makes two-component calculations less computationally expensive than four-component calculations is that the small component is ignored and not calculated. Another popular approximation that makes two-component calculation less computationally expensive is that only terms involving one electron are used. For two-component calculations, it is assumed that there is a matrix X that satisfies the following equation. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1GSjYlUSJTRidGTUZQRi8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmhuLUZHNiUtRko2JVEnWCZwc2k7RidGTUZQLUYjNiQtRko2JVEiTEYnRk1GUEYvRldGKy8lK2V4ZWN1dGFibGVHRjRGLw== This gives the unitary operator the following form. 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 This unitary transform can be applied to the Dirac equation along with requiring that off diagonal elements are zero to give the equation for two-component calculations. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEqY1gmc2lnbWE7RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RJyZzZG90O0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRkc2JVEjcFhGJ0ZKRk0tRiw2LVEqJnVtaW51czA7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRlktRkc2JVEpYyZzaWdtYTtGJ0ZKRk1GTy1GRzYlUSJwRidGSkZNLUYsNi1RIitGJ0YvRjJGNUY3RjlGO0Y9Rj9GWEZaLUkobWZlbmNlZEdGJDYmLUYjNiYtRkc2JVEiWEYnRkpGTS1GLDYtUSIsRidGL0YyL0Y2RkxGN0Y5RjtGPUY/RkEvRkVRLDAuMzMzMzMzM2VtRictRkc2JVEiVkYnRkpGTUYvRi8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGW28tSSNtbkdGJDYkUSIyRidGLy1JJW1zdXBHRiQ2JS1GRzYlUSNtY0YnRkpGTS1GIzYkRmVwRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmNvLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGaHEtRmZwNiRGY3FGLy8lK2V4ZWN1dGFibGVHRjRGLw== The difficult part of two component calculations is that the exact form of X is unknown. There are many different methods that assume various forms of X, such as X2C and DKH methods, but this worksheet will not go into further detail on these methods. The drive to find ways to practically implement relativity into quantum mechanical calculations has resulted in a variety of methods that vary in their complexity and their accuracy. These methods and the concepts that have led up to them can now be used to discuss the surprising importance relativity has on lead-acid batteries.

\342\200\234Relativity and the lead-Acid Battery\342\200\235 by Rajeev Ahuja et al, took a deep look at the lead acid battery and how relativistic effects play a major role on its voltage. They found that about \342\200\2341.7-1.8V\342\200\235 of the \342\200\2342.13V\342\200\235 produced by lead-acid batteries come from the relativistic effects in the lead atoms [2]. Ahuja et al suspect that the relativistic corrections to the 6s orbital play a significant role in the dramatic increase in the voltage of lead-acid batteries. Going through and recalculating all of their results would be beyond the scope of this worksheet, but this worksheet will calculate the energy of the 6s orbital using a non-relativistic method and compare the value to an experimental value. The following is a calculation of the energy of the 6s orbital of a lead atom using the Hartree-Fock method with the \342\200\234dzp\342\200\235 basis set. 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnRGlnaXRzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJkFzc2lnbjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzE1RidGOUY5

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The Hartree Fock method calculates the 6s orbital of Pb to have an energy of about -12.56 eV. Relativistic methods calculate the energy of the 6s orbital of Pb to be -15.42 eV [3]. These two values are in line with the values used in \342\200\234Relativity and the lead-Acid Battery\342\200\235 [2]. This is a change of 22.77% which is enough to have significant consequences, such as being responsible for the vast majority of the voltage of lead acid batteries. For heavy atoms, relativistic effects play a major role in the orbital energies and the physical and chemical properties of these atoms.

The difference in the calculated energy of the 6S orbital of lead between the Hartree-Fock method and the relativistic method indicates that the two models of the orbitals differ significantly. There is more to the difference of the models besides just the energies. As mentioned previously the mass of the electrons differ according to the following relationship. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYjNictSSNtaUdGJDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGVS1JJW1zdWJHRiQ2JUZILUYjNiQtSSNtbkdGJDYkUSIwRidGL0YvLyUvc3Vic2NyaXB0c2hpZnRHRmluLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtc3FydEdGJDYjLUYjNiYtRmduNiRRIjFGJ0YvLUYsNi1RKCZtaW51cztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGYHAtSSZtZnJhY0dGJDYoLUZdbzYlLUZJNiVRInZGJ0ZMRk8tRiM2JC1GZ242JFEiMkYnRi9GLy8lMXN1cGVyc2NyaXB0c2hpZnRHRmluLUYjNiQtRl1vNiUtRkk2JVEiY0YnRkxGT0ZqcEZfcUYvLyUubGluZXRoaWNrbmVzc0dGW3AvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXHIvJSliZXZlbGxlZEdGNEYvRi9GLy1GIzYlLUYsNi1RKiZ1bWludXMwO0YnRi9GMkY1RjdGOUY7Rj1GP0ZfcEZhcEZpb0YvRl9xRi8tRkk2I1EhRicvJStleGVjdXRhYmxlR0Y0Ri8= The orbitals also have a different size. \342\200\234Outermost s spinors given by the Dirac-Fock calculations are contracted compared to those predicted by nonrelativistic Hartree-Fock calculations\342\200\235 and \342\200\234outermost np+ spinors are more diffuse than the corresponding np orbital\342\200\235 [4]. These physical consequences of relativity mean that the 6s spinor of lead is smaller in size, lower in energy, and holds a heavier electron than the 6s orbital of lead. Relativistic quantum chemistry has been used to model many systems beyond the lead-acid battery. Relativistic calculations have found that the voltage of mercury acid batteries is about \342\200\23429%\342\200\235 due to relativistic effects [5]. Beyond batteries, relativistic calculations have also been done to model the different ionization energies of transition metals while bounded to different ligands [6]. Relativity is an essential aspect needed to model any system that includes heavy atoms. The Dirac equation does a great job at modeling relativistic quantum systems, but it is not a perfect theory. Some of the problems of the Dirac equation include an Asymmetry between electrons and positrons, it is not applicable to spin-0 particles, and it assumes that negative-energy electrons do not have a potential that acts on the positive-energy electrons [7]. These problems along with other problems not mentioned lead to a stronger theory known and quantum electrodynamics (QED). Molecular QED is a theory that is used when extremely high precision calculations are needed [7]. The calculations performed in this worksheet found the orbital energy of the 6s orbital of lead to be -12.56 eV which is 22.77% off the relativistic calculation. This massive discrepancy highlights that Hartree-Fock theory is missing some of the essential properties of real systems. By noticing that core electrons of heavy atoms move at speeds close to the speed of light, it is obvious that what Hartree-Fock is missing is relativity. Relativity can be incorporated into quantum mechanics by using the Dirac equation. The Dirac equation allows for better modeling of systems that include heavy atoms. The improved accuracy of calculations performed with the Dirac equation should make it clear to any researcher who is performing computational work on batteries or any system involving heavy atoms that relativity must be considered. Relativity must be accounted for alongside basis sets, and electron correlation as the three pillars that determine the accuracy of a quantum chemistry calculation.

1.) Jensen, F. Introduction to computational chemistry; John Wiley & Sons Inc: Hoboken, NJ, 2017. 2.) Ahuja, R.; Blomqvist, A.; Larsson, P.; Pyykk\303\266, P.; Zaleski-Ejgierd, P. Physical Review Letters 2011, 106 (1). 3.) Desclaux, J. P. Atomic Data and Nuclear Data Tables 1973, 12 (4), 311\342\200\223406. 4.) Tatewaski, H.; Yamamoto, S.; Hatano, Y. Relativistic Effects in the Electronic Structure of Atoms. ACS Omega 2017. 5.) Zaleski-Ejgierd, P.; Pyykk\303\266, P. Physical Chemistry Chemical Physics 2011, 13 (37), 16510.6.) Schwarz, H. ChemInform 2003, 34 (51). 7.) Liu, W. Essentials of Relativistic Quantum Chemistry. The Journal of Chemical Physics 2020, 152 (18), 180901.