Generalized spin mapping is easiest to read as a geometry correction to MMST mapping. Instead of first embedding electronic states into fictitious harmonic oscillators, Runeson and Richardson start from the symmetry of the finite electronic Hilbert space itself. The essential objects are the \(SU(N)\) generators, the quadratic Casimir invariant, the physical pure-state manifold, and the Stratonovich-Weyl transform for finite-dimensional phase space.

This note supplies the mathematical layer for the companion article Spin Mapping and Spin-MInt Method, where the same \(SU(N)\) structure is turned into a direct Spin-MInt propagator and tested against FSSH, Ehrenfest, and exact DVR references.

Background

In nonadiabatic dynamics, a molecule is often split into classical nuclear coordinates and a finite electronic subsystem. If there are \(N\) diabatic electronic states, the electronic Hamiltonian is an \(N\times N\) Hermitian matrix. MMST mapping handles this by assigning one fictitious oscillator to each electronic state. That produces useful continuous variables, but it also enlarges the electronic phase space beyond the physical singly-excited oscillator subspace.

The appearance of \(SU(N)\) is therefore a direct consequence of finite-state electronic quantum mechanics, not an additional physical force. After truncation to \(N\) electronic states, the wavefunction coefficients form a normalized vector \(\mathbf c(t)\in\mathbb C^N\) propagated by \(i\hbar\dot{\mathbf c}=H_e\mathbf c\), so a Hermitian electronic Hamiltonian generates a unitary \(U(N)\) evolution. The determinant phase, equivalently the trace term \(h_0 I\) in \(H_e=h_0I+\sum_a h_a S_a\), only multiplies the state by a global phase and leaves populations, coherences, and density matrices unchanged. The physically relevant coherent rotation is generated by the traceless part \(\sum_a h_a S_a\), with Pauli generators for \(N=2\), Gell-Mann generators for \(N=3\), and generalized \(SU(N)\) generators for larger electronic spaces.

Spin mapping asks a different question: what continuous phase space belongs to the original \(N\)-level electronic system before any oscillator embedding is introduced? For two states, the answer is familiar: the system is equivalent to a spin-\(1/2\) particle on a Bloch sphere. For \(N\) states, the analogous symmetry is \(SU(N)\).

The SU(N) Algebra

The special unitary group is

\[ SU(N)=\{U\in\mathbb C^{N\times N}\mid U^\dagger U=I,\ \det U=1\}. \]

The global phase is removed from the physical electronic state, so the nontrivial rotations among \(N\) states are described by \(SU(N)\), not by the full \(U(N)\). Its Lie algebra has \(N^2-1\) traceless Hermitian generators, denoted here by \(\hat S_i\), with the normalization

\[ \mathrm{tr}(\hat S_i\hat S_j)=\frac{1}{2}\delta_{ij}, \qquad i,j=1,\ldots,N^2-1. \]

For \(N=2\), these are \(\sigma_i/2\), the Pauli matrices divided by two. For \(N=3\), they are the Gell-Mann matrices divided by two. A general electronic operator can then be decomposed as

\[ \hat A=A_0\hat I+\sum_{i=1}^{N^2-1} A_i\hat S_i, \qquad A_0=\frac{1}{N}\mathrm{tr}(\hat A). \]

This split is not a numerical convention. The identity carries the trace part, while the \(SU(N)\) generators carry the traceless part. That is why the spin-mapping construction is insensitive to arbitrary choices of how a potential matrix is separated into state-independent and state-dependent pieces.

Casimir Invariant

The key invariant is the quadratic Casimir,

\[ \sum_{i=1}^{N^2-1}\hat S_i^2 = \frac{N^2-1}{2N}\hat I. \]

This is the \(SU(N)\) version of total spin length. For ordinary spin angular momentum, \(\hat S_x^2+\hat S_y^2+\hat S_z^2=S(S+1)\). With \(N=2\), the formula gives

\[ \frac{N^2-1}{2N}=\frac{3}{4}, \]

which is exactly the spin-\(1/2\) value. In generalized spin mapping, this invariant fixes the natural length scale of the electronic phase-space variables. This is where the zero-point-energy parameter in the W-representation comes from:

\[ \gamma_W=\frac{2}{N}\left(\sqrt{N+1}-1\right). \]

For the seven-state FMO model used by Runeson and Richardson, this gives \(\gamma_W\approx0.522\), much closer to the reduced values often found empirically useful in MMST-based calculations than to the original MMST choice \(\gamma=1\).

Physical Versus Algebraic Degrees of Freedom

A common trap is to equate the \(N^2-1\) generators with \(N^2-1\) independent pure-state coordinates. That is only safe for \(N=2\). A normalized electronic pure state is

\[ |\Omega\rangle=\sum_{n=1}^{N} c_n|n\rangle, \qquad \sum_{n=1}^N |c_n|^2=1. \]

The \(N\) complex coefficients contain \(2N\) real variables. Normalization removes one. The global phase removes another. Therefore a pure \(N\)-level electronic state has

\[ 2N-2 \]

real physical degrees of freedom. The pure-state manifold is the complex projective space \(\mathbb{CP}^{N-1}\).

The generalized Bloch vector has \(N^2-1\) components, \(\langle\Omega|\hat S_i|\Omega\rangle\), but for \(N>2\) those components obey constraints and do not fill a full \((N^2-1)\)-dimensional sphere. This distinction matters for mapping dynamics: a physical spin mapping should preserve the constrained \(2N-2\)-dimensional electronic geometry, not freely propagate all \(N^2-1\) algebra components as independent coordinates.

Why Not Ordinary Wigner?

The ordinary Wigner transform is built for continuous canonical variables. For a nuclear coordinate, it has the familiar form

\[ A_W(q,p)=\int d\xi\, e^{ip\xi/\hbar} \left\langle q-\frac{\xi}{2}\middle|\hat A\middle|q+\frac{\xi}{2}\right\rangle. \]

This formula assumes a continuous coordinate \(q\), a conjugate momentum \(p\), a flat phase space, and the Heisenberg-Weyl translation structure. Nuclear motion satisfies those assumptions well enough for semiclassical dynamics. A finite electronic subsystem does not. It has discrete basis states \(|1\rangle,\ldots,|N\rangle\), but no native continuous \(q,p\) pair.

MMST makes ordinary phase-space ideas available by embedding the electronic states into harmonic oscillators. That is practical, but it is also the source of the enlarged unbounded mapping space and the physical-subspace leakage problem. The Stratonovich-Weyl transform avoids this detour by defining a Wigner-like representation directly on the finite \(SU(N)\) coherent-state space.

Stratonovich-Weyl Transform

For a finite electronic subsystem, the Stratonovich-Weyl transform maps an operator to a function on the coherent-state phase space:

\[ A_s(\Omega)=\mathrm{tr}[\hat A\hat w_s(\Omega)]. \]

The index \(s\) labels \(Q\), \(P\), or \(W\) representations. With the normalization used above, the kernels are

\[ \hat w_Q(\Omega)=\frac{1}{N}\hat I +2\sum_{i=1}^{N^2-1} \langle\Omega|\hat S_i|\Omega\rangle\hat S_i, \]
\[ \hat w_P(\Omega)=\frac{1}{N}\hat I +2(N+1)\sum_{i=1}^{N^2-1} \langle\Omega|\hat S_i|\Omega\rangle\hat S_i, \]
\[ \hat w_W(\Omega)=\frac{1}{N}\hat I +2\sqrt{N+1}\sum_{i=1}^{N^2-1} \langle\Omega|\hat S_i|\Omega\rangle\hat S_i. \]

The W-representation is especially useful because it is self-dual:

\[ \mathrm{tr}(\hat A\hat B) = \int d\Omega\, A_W(\Omega)B_W(\Omega). \]

That property mirrors the role of the ordinary Wigner transform for continuous variables. The right summary is therefore not that Wigner is wrong, but that ordinary Wigner belongs to canonical nuclear coordinates, while Stratonovich-Weyl is the finite-dimensional \(SU(N)\) analogue used for electronic states.

Mapping Variables

To propagate trajectories, the coherent-state coefficients are rewritten as Cartesian variables:

\[ (N+1)^{1/4}c_n=\frac{X_n+iP_n}{\sqrt{2}}. \]

Normalization gives the W-representation hypersphere constraint

\[ \sum_{n=1}^N(X_n^2+P_n^2)=2\sqrt{N+1}. \]

The resulting W-representation Hamiltonian has the same practical shape as MMST,

\[ H_W = \frac{p^2}{2m} + \sum_n V_n(x)\frac{1}{2}(X_n^2+P_n^2-\gamma_W) + \sum_{n>m}V_{nm}(x)(X_nX_m+P_nP_m), \]

but the geometry is different. The variables are sampled on a fixed-radius electronic phase space, the identity maps to one, and \(\gamma_W\) is fixed by the \(SU(N)\) construction rather than tuned as an empirical parameter.

Diagnostic Check

The following script constructs generalized Gell-Mann generators and checks the two algebraic identities used above:

  • su_n_spin_mapping_checks.py verifies generator orthogonality, the quadratic Casimir, the pure-state degree count, and the W-representation \(\gamma_W\) values.

Running the script from the repository root gives:

N  generators  pure_dof  bloch_components  Casimir       gamma_W      R_W^2        max_errors
2  3          2         3                 0.75000000   0.73205081   3.46410162   0.00e+00
3  8          4         8                 1.33333333   0.66666667   4.00000000   2.22e-16
7  48         12        48                3.42857143   0.52240775   5.65685425   4.44e-16
8  63         14        63                3.93750000   0.50000000   6.00000000   4.44e-16

The useful comparison is between pure_dof and bloch_components. For \(N=7\), the electronic pure state has only 12 physical real degrees of freedom, even though its generalized Bloch representation has 48 algebra components. The small numerical errors confirm the expected normalization and Casimir identities.

Takeaway

The mathematical role of \(SU(N)\) in spin mapping is to keep the electronic subsystem on its own finite-dimensional geometry. The Casimir invariant supplies the generalized spin length. The difference between \(2N-2\) physical pure-state degrees of freedom and \(N^2-1\) algebra components explains why the phase space is constrained. The Stratonovich-Weyl transform then provides the finite-state analogue of a Wigner representation, allowing electronic operators and correlation functions to be evaluated without embedding the electronic problem into an unbounded oscillator space.

References

  1. J. E. Runeson and J. O. Richardson, "Generalized spin mapping for quantum-classical dynamics," Journal of Chemical Physics 152, 084110 (2020).
  2. J. E. Runeson and J. O. Richardson, "Spin-mapping approach for nonadiabatic molecular dynamics," Journal of Chemical Physics 151, 044119 (2019).