Coordinate transformations Slater determinants

First, how do the particles move in VMC In the continuum it is usually more efficient to move the particles one at a time by adding a random vector to a particle s coordinate, where the vector is either uniform inside of a cube, or is a normally distributed random vector centered around the old position. Unfortunately, this procedure cannot be used with backflow orbitals. This is because the backflow transformation couples all the electronic coordinates in the orbitals so that once a single electron move is attempted the entire Slater determinant needs to be recomputed, an 0 N ) operation. It is much more efficient to move all electrons at once, although global moves could become inefficient for large systems. 

We will need the unitary transformations exp (lA) and exp (iS). They are very convenient, since when starting from some set of the orthonormal functions (spinorbitals or Slater determinants) and applying this transformation, we always retain the orthonormality of new spinorbitals (due to A) and of the linear combination of determinants (due to S). This is an analogy to the rotation of the Cartesian coordinate system. It follows from the above equations that exp (/A) modifies spinorbitals (i.e., operates in the one-electron space), and exp (i S) rotates the determinants in the space of many-electron functions. 

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