... nuclei^2.1

At the atomic energy scales which are of interest in this work, the nuclei are extremely well-described as massive point charges and their internal structure is safely neglected.

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... particle^2.2

We are neglecting spin in this discussion.

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... eigenvalue^2.3

For the case of eigenvalue degeneracy, the state-vector collapses to a vector lying in the subspace spanned by all of the eigenvectors corresponding to the measured eigenvalue.

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... measurement^2.4

Again, for the degenerate case, the probabilities must be summed for all eigenvectors corresponding to the measured eigenvalue.

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... operators^2.5

For a linear operator ${\hat O}$, ${\hat O} (\alpha \vert A \rangle + \beta \vert B \rangle ) = \alpha {\hat O} \vert A \rangle + \beta {\hat O} \vert B \rangle$.

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... equation^2.6

Atomic units are used throughout (unless otherwise stated): $\hbar = m_{\mathrm e} = e = 4 \pi \varepsilon_0 = 1$.

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... problem^2.7

The most notable exception to this rule is the motion of hydrogen, which is often treated using the path-integral formulation of quantum mechanics[9,10].

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... express^2.8

Berry [13] has recently proposed a non-relativistic explanation involving a geometric phase.

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... operators^2.9

Neither the creation nor annihilation operators are Hermitian, and so they do not correspond to physical observables.

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... states^3.1

Our restriction to non-spin-polarised systems requires that $N$ be even.

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... potential^3.2

The chemical hardness has been proposed as a quantity which gives a more reliable indication of pseudopotential transferability since it includes self-consistent effects [63,64].

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... density-matrix^4.1

The computational cost of diagonalising a sparse density-matrix scales as the square of the system-size.

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... reciprocal-space^7.1

The conventions used here for discrete Fourier transforms are

$${\tilde n}({\bf G}) = \sum_{\bf r} n({\bf r}) \exp(-{\mathrm ... ...} {\tilde n}({\bf G}) \exp({\mathrm i} {\bf G} \cdot {\bf r}).$$

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