Cartesian harmonic oscillator basis

One often uses the Cartesian harmonic oscillator (HO) basis to solve the self-consistent equations when neither spherical nor axial symmetry is assumed, see, e.g., Refs. [17,10]. The Cartesian HO states are identified by the numbers of oscillator quanta, n_x, n_y, and n_z, in the three Cartesian directions, and by the spin projection s_z= $\pm\frac{1}{2}$ on the z axis. For the standard HO phase convention, this basis is real, $\hat{\cal{K}}\vert n_xn_yn_z,s_z\rangle=\vert n_xn_yn_z,s_z\rangle$, where according to our standard convention the script symbol $\hat{\cal{K}}$ denotes the coordinate-space complex-conjugation operator acting in the odd-fermion-number space ${\cal{H}}_-$, see Sec. 2.1.

For the HO states the following relations hold [18]:

       

$$\hat{\cal{P}}\vert n_xn_yn_z,s_z\rangle \phantom{i}(-1)^{n_x+n_y+n_z}\vert n_xn_yn_z,s_z\rangle ,$$ = (47)

$$\hat{\cal{T}}\vert n_xn_yn_z,s_z\rangle \phantom{i}(-1)^{\frac{1}{2}-s_z}\vert n_xn_yn_z,-s_z\rangle ,$$ = (48)

$$\hat{\cal{R}}_{x}\vert n_xn_yn_z,s_z\rangle i(-1)^{n_y+n_z+1}\vert n_xn_yn_z,-s_z\rangle ,$$ = (49)

$$\hat{\cal{R}}_{y}\vert n_xn_yn_z,s_z\rangle \! \!\! \phantom{i}(-1)^{n_x+n_z+\frac{1}{2}-s_z}\vert n_xn_yn_z,-s_z\rangle ,$$ = (50)

$$\hat{\cal{R}}_{z}\vert n_xn_yn_z,s_z\rangle i(-1)^{n_x+n_y+\frac{1}{2}+s_z}\vert n_xn_yn_z,s_z\rangle ,$$ = (51)

$$\bar{\cal{E}}\vert n_xn_yn_z,s_z\rangle - \vert n_xn_yn_z,s_z\rangle ,$$ = (52)

from where one can find similar equations for all the remaining operators of group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$. Since the HO Hamiltonian is symmetric under D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, its eigenstates can be classified according to the ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$. It is easily seen that the HO states $\vert n_xn_yn_z,s_z\rangle$ form bases of the spinor, pseudospinor, antispinor and antipseudospinor ircoreps for $\{N$=n_x+n_y+n_z, N_y=n_x+$n_z\}$ being {even, odd}, {odd, odd}, {even, even} and {odd, even}, respectively (see Appendix). The entire HO basis would have belonged to the spinor and pseudospinor ircoreps only, if the basis states and phase convention were chosen differently, see Ref.[14].