The unimodular-like orthogonal O$_{(}$$_{S}$$_{)}$ ( N$_{c}$ ) symmetry group is constructed by imposing the unimodular-like constraint ∑ N$_{c}$ i $_{=1}$θ$_{i}$ = 0 to the standard orthogonal symmetry group. The color-singlet Gaussian integral for the unimodular-like orthogonal symmetry group S O$_{(}$$_{S}$$_{)}$ ( N ) is related to O$_{(}$$_{S}$$_{)}$ ( N ) in the following way C. Symplectic symmetry group Sp ( N ) (N is the number of quaternion) The symplectic symmetry group is time-reversal invariance. The anti-symmetrization in the symmetry group Sp ( N ) leads to odd-spin. The algebra of the symplectic symmetry group can be expressed in terms of quaternions. The quaternions are convenient to deal with the anti-symmetrization. The notation Sp ( N ) is equivalent to Sp (2 N, R ) where N corresponds to the number of quaternions. The transformation matrix U of the Sp ( N ) group is of the size of (2 N × 2 N ). This transformation matrix is considered as cut into N 2 blocks of 2 × 2 and each 2 × 2 block is expressed in terms of the quaternions. In this sense the number of real independent parameters these are defined in the ( 2 N × 2 N ) self-dual Hermitian matrix is given by the number of the adjoint variables N$_{adj}$ . The number of the fundamental (energy-) eigen variables is N$_{fun}$ = N while the number of the adjoint variables is N$_{adj}$ = N (2 N − 1). The invariance Haar measure for the symplectic symmetry group Sp ( N ) with the circular canonical ensemble is given by The normalisation constant N$_{Sp}$ is determined by Under extreme conditions, the saddle points of the group continuous variables accumulate around the origin. Therefore, the canonical ensemble is approximated to the Gaussian-like 24 1 N$_{Sp}$ = ∫ π − π N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j 16 sin 4 ( θ$_{i}$ − θ$_{j}$ 2 ) . (78) ∫ dµ$_{Sp}$ ( θ$_{1}$ · · · θ$_{N}$ ) = N$_{Sp}$ ∫ π − π N ∏ i =1 dθ$_{i}$ 2 π ∏ i<j [ 4 ( e iθ$_{i}$ − e iθ$_{j}$ ) ( e iθ$_{i}$ − e iθ$_{j}$ ) ∗ ] $^{2}$. (77) ∫ dµ$_{O}$$_{(}$$_{S}$$_{)}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i , (with the unimodular-like constraint) , ∼ ∫ dµ$_{O}$$_{(}$$_{S}$$_{)}$ ( θ$_{1}$ · · · θ$_{N}$ ) 2 π δ ( N ∑ i =1 θ$_{i}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i ∼ √ 2 π a N ∫ dµ$_{O}$$_{(}$$_{S}$$_{)}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i . (76)