<pre><span>The</span><span> total </span><span>energy</span><span> </span><span>is</span><span> a </span><span>fundamental</span><span> </span><span>characteristic</span><span> </span><span>of</span><span> </span><span>solids</span><span>, </span><span>molecules</span><span>, and </span><span>nanostructures</span><span>. </span><span>In </span><span>most</span><span> </span><span>first</span><span>-</span><span>principles</span><span> </span><span>calculations</span><span> </span><span>of</span><span> </span><span>the</span><span> total </span><span>energy</span><span>, </span><span>the</span><span> </span><span>nuclear</span><span> </span><span>kinetic</span><span> operator </span><span>is</span><span> </span><span>decoupled</span><span> from </span><span>the</span><span> </span><span>many</span><span>-body </span><span>electronic</span><span> </span><span>Hamiltonian</span><span> and </span><span>nuclear</span><span> </span><span>potential</span><span>, and </span><span>the</span><span> </span><span>dynamics</span><span> </span><span>of</span><span> </span><span>the</span><span> </span><span>nuclei</span><span> </span><span>is</span><span> </span><span>reintroduced</span><span> </span><span>afterward</span><span>. </span><span>This</span><span> </span><span>two</span><span>-step </span><span>procedure</span><span> </span><span>introduced</span><span> </span><span>by</span><span> </span><span>Born</span><span> and Oppenheimer (</span><span>BO</span><span>) </span><span>is</span><span> </span><span>approximate</span><span>. </span><span>Energies</span><span> </span><span>beyond</span><span> </span><span>the</span><span> </span><span>electronic</span><span> and </span><span>vibrational</span><span> (or </span><span>phononic</span><span>) main contributions </span><span>might</span><span> be relevant </span><span>when</span><span> small </span><span>energy</span><span> </span><span>differences</span><span> are important, </span><span>such</span><span> as </span><span>when</span><span> </span><span>predicting</span><span> stable </span><span>polymorphs</span><span> or </span><span>describing</span><span> </span><span>the</span><span> </span><span>magnetic</span><span> </span><span>energy</span><span> landscape. </span><span>We</span><span> </span><span>clarify</span><span> </span><span>the</span><span> </span><span>different</span><span> </span><span>flavors</span><span> </span><span>of</span><span> </span><span>BO</span><span> </span><span>decoupling</span><span> and </span><span>give</span><span> an exact formulation for </span><span>the</span><span> total </span><span>energy</span><span> in </span><span>the</span><span> </span><span>basis</span><span> </span><span>of</span><span> </span><span>BO</span><span> </span><span>electronic</span><span> </span><span>wavefunctions</span><span>. </span><span>Then</span><span>, </span><span>we</span><span> list contributions </span><span>beyond</span><span> </span><span>the</span><span> main </span><span>ones</span><span> </span><span>that</span><span> </span><span>appear</span><span> in a perturbative expansion in </span><span>powers</span><span> </span><span>of</span><span> </span>M<sub>0</sub><sup>-1/4</sup><span>, </span><span>where</span><span> </span>M<sub>0</sub><span> </span><span>is</span><span> a </span><span>typical</span><span> </span><span>nuclear</span><span> </span><span>mass</span><span>, up </span><span>to</span><span> </span><span>sixth</span><span> </span><span>order</span><span>. </span><span>Some</span><span> </span><span>of</span><span> </span><span>these</span><span> </span><span>might</span><span> be </span><span>grouped</span><span> and </span><span>denoted</span><span> as </span><span>the</span><span> </span><span>electron</span><span>-phonon contribution </span><span>to</span><span> total </span><span>energy</span><span>, </span><span>$E^{\mathrm{elph}}$</span><span>, </span><span>which</span><span> </span><span>first</span><span> </span><span>appears</span><span> </span><span>at</span><span> </span><span>fourth</span><span> </span><span>order</span><span>. </span><span>The</span><span> </span><span>electronic</span><span> </span><span>inertial</span><span> </span><span>mass</span><span> </span><span>contributes</span><span> </span><span>at</span><span> </span><span>sixth</span><span> </span><span>order</span><span>. </span><span>We</span><span> </span><span>clarify</span><span> </span><span>that</span><span> </span><span>the</span><span> sum </span><span>of</span><span> </span><span>the</span><span> </span><span>Allen</span><span>-</span><span>Heine</span><span>-</span><span>Cardona</span><span> zero-point </span><span>renormalization</span><span> </span><span>of</span><span> </span><span>eigenvalues</span><span> </span><span>over</span><span> </span><span>occupied</span><span> states </span><span>is</span><span> not </span><span>the</span><span> </span><span>electron</span><span>-phonon contribution </span><span>to</span><span> </span><span>the</span><span> total </span><span>energy</span><span> but </span><span>rather</span><span> part </span><span>of</span><span> </span><span>the</span><span> </span><span>phononic</span><span> contribution. </span><span>The</span><span> computation </span><span>of</span><span> </span><span>the</span><span> </span><span>lowest</span><span>-</span><span>order </span>E<sup>elph</sup><span> </span><span>is</span><span> </span><span>implemented</span><span> and </span><span>shown</span><span> </span><span>to</span><span> be small but non-</span><span>negligible</span><span> in </span><span>the</span><span> case </span><span>of</span><span> diamond (3.8~meV </span><span>per</span><span> </span><span>atom</span><span>) and </span><span>GaN</span><span> (9.7~meV </span><span>per</span><span> </span><span>atom</span><span>).</span></pre>
<pre><span>We</span><span> also </span><span>estimate</span><span> </span><span>the</span><span> </span><span>electronic</span><span> </span><span>inertial</span><span> </span><span>mass</span><span> contribution and </span><span>the</span><span> quasi-</span><span>harmonic</span><span> </span><span>one</span><span> for diamond. </span><span>We</span><span> </span><span>confirm</span><span> </span><span>the</span><span> size </span><span>consistency</span><span> </span><span>of</span><span> all </span><span>computed</span><span> </span><span>terms</span><span>.</span></pre>
