... nuclei2.1
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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... particle2.2
We are neglecting spin in this discussion.
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... eigenvalue2.3
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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... measurement2.4
Again, for the degenerate case, the probabilities must be summed for all eigenvectors corresponding to the measured eigenvalue.
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... operators2.5
For a linear operator , .
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... equation2.6
Atomic units are used throughout (unless otherwise stated): .
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... problem2.7
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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... express2.8
Berry [13] has recently proposed a non-relativistic explanation involving a geometric phase.
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... operators2.9
Neither the creation nor annihilation operators are Hermitian, and so they do not correspond to physical observables.
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... states3.1
Our restriction to non-spin-polarised systems requires that be even.
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... potential3.2
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-matrix4.1
The computational cost of diagonalising a sparse density-matrix scales as the square of the system-size.
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... reciprocal-space7.1
The conventions used here for discrete Fourier transforms are

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