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Sometimes the concept "manifold" is used in quantum chemistry, for instance, in Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function:

where $\tau_{2s}$ and $\tau_{2t}$ denote the singlet-singlet and triplet-triplet spin coupled double excitation manifolds where spin couplings initially are carried out on the occupied-occupied and unoccupied-unoccupied orbital indices.

However, in general, manifold refers to a geometric object that is locally Euclidian. This is not essential to the coupled cluster method.

My question is, what is the motivation to introduce manifolds in the coupled cluster method?

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    $\begingroup$ In general, the term "manifold" might refer to about a dozen of different and unrelated things (this, for example). Using it without any context is a great way to get misunderstood. $\endgroup$ – Ivan Neretin Nov 1 '16 at 15:57
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The use of the word "manifold" in most of the quantum chemistry literature is purely as a synonym for multiple or different. Usually it appears when the number may be very large. Nothing more complicated than that.

For a simpler example than coupled cluster, consider configuration interaction with only single excitations (CIS). One of the key quantities is the set of single excitation amplitudes, $t_{ia}$ or $t_{i}^{a}$ or $k_{ai}\hat{a}^{\dagger}\hat{i}$ depending on your preference for notation. $i$ is an occupied MO index, and $a$ is a virtual MO index. We might say that the set $\{t\}$ forms a manifold, and the number quickly becomes large as you increase the basis set size: there will be $N_{ov} = N_{\text{occ}} \times N_{\text{virt}}$ single excitation amplitudes.

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