The canonical ensemble with specific internal symmetry group and that is projected to a certain quantum wavefuntion is given by where Z$_{0}$ ( β, V ) = exp ( a (0) q ¯ qg ) . The symmetry group structure is defined by G = 1, 2, 4 for the orthogonal, unitary and symplectic ensemble, respectively. The color-singlet-state quantum wave-function is given by χ$_{singlet}$ = 1 regardless of the symmetry group. In general the physical state in QCD is assumed to satisfy the condition det( G ) = 1 or equivalently it is restricted to the unimodular-like constraint, namely, ∑ N$_{c}$ i =1 θ$_{i}$ = 0. The unimodular-like constraint modifies the extrapolated color-singlet-state canonical Gaussian ensemble with a specific group such as unitary, orthogonal and symplectic symmetry groups to the following quantity where N = N$_{U}$ , N$_{O}$ and N$_{Sp}$ for the unitary, orthogonal and symplectic symmetry groups, respectively. Eq.(84) is derived by imposing the unimodular-like constraint using the fol- lowing trick The integral part in Eq.(84) is evaluated as follows The exponent α is determined by 26 α = G ( N 2 c − N$_{c}$ ) / 4 + N$_{c}$/ 2 . (87) I = ∫ ( N$_{c}$ ∏ n =1 dθ$_{n}$ ) N$_{c}$ ∏ j>i | θ$_{j}$ − θ$_{i}$ | G e − 1 $_{2}$a (2) qqg ∑ N c i $_{=1}$θ 2 i , = ( G a (2) $_{qqg}$) G ( N 2 $_{c}$− N$_{c}$ ) / 4+ N$_{c}$/ $^{2}$∫ ∞ −∞ ( N$_{c}$ ∏ n =1 dx$_{n}$ ) ∏ j>i | x$_{j}$ − x$_{i}$ | G e − 1 2 G ∑ N c i x 2 i , = N N$_{c}$ G ( G a (2) $_{qqg}$) α . (86) δ ( N ∑ i =1 θ$_{i}$ ) = ∫ ∞ −∞ dx e ix ∑ N 1 $^{θ$_{i}$}$. (85) Z$_{singlet}$ ( β, V ) ∼ N Z$_{0}$ ( β, V ) √ 2 π a (2) qqg N$_{c}$ ∫ ( N$_{c}$ ∏ n =1 dθ$_{n}$ ) N$_{c}$ ∏ j>i | θ$_{j}$ − θ$_{i}$ | G × e − 1 $_{2}$a (2) qqg ∑ N c i $_{=1}$θ 2 i , (84) Z$_{state}$ ( β, V ) = ∫ dµ$_{G}$ ( g ) [ χ$_{state}$ × Z$_{q}$$_{¯}$ qg ( β, V ; θ$_{1}$, · · · , θ$_{N}$ )] , ∼ N Z$_{0}$ ( β, V ) ∫ ( N$_{c}$ ∏ n =1 dθ$_{n}$ ) ∏ j>i | θ$_{j}$ − θ$_{i}$ | G ( χ$_{state}$ ) e − 1 $_{2}$a (2) qqg ∑ N c i θ 2 i , (83)