Λ$_{λ}$ ( x | x $^{′}$) = x x ′ − λ x ′ x 1 1 ... x ′ 3 2 1 + − x 3 1 2 $_{λ}$3 2 − x ′ x 1 1 $_{λ}$2 2 2 2 − − x ′ x 2 2 $_{λ}$3 6 3 3 1 1 4 FIG. 2: The skeleton diagrams for correlation function Λ λ . Double dashed lines depict the tunneling Green’s functions T λ and the rectangles depict the irreducible Green’s functions. III. THERMODYNAMIC POTENTIAL DIAGRAMS In expression (17) the coefficients 1 n before each dia- gram are absent, where n is the order of perturbation theory. These coefficients are present in Fig. 3. In order to restore these 1 n coefficients in (17) and obtain the co- incidence with 〈 U$_{λ}$ ( β ) 〉 c 0 series it is enough to integrate by λ the expression (17) and obtain: The thermodynamic potential of the system is deter- mined by the connected part of the mean value of the evolution operator [11 , 12] Let us consider a more general quantity first The expression (18) displayed in a diagrammatic repre- sentation coincides exactly with mean value of the evo- lution operator: and put then λ = 1. By using the perturbation theory we have obtained the first orders of diagrams for 〈 U$_{λ}$ ( β ) 〉 c $_{0}$, depicted in Fig. 3. In order to obtain the better understanding of these diagrammatic contributions we examine the expression In Fourier representation we have where double repeated indices suppose summation and integration. Consequently (16) is equal to From (15) and (20) we obtain Here we have carried out the integration by time. From diagrammatic point of view equation (16) implies the procedure of locking of the external lines of propaga- tors G$_{λ}$ diagrams depicted on the Fig. 1 with the tun- neling matrix element t ( − → x ′ − − → x ) and obtaining in such a way the diagrams without external lines similar with ones for 〈 U$_{λ}$ ( β ) 〉 c 0 depicted on Fig. 3. These two series of diagrams differ by coefficients in front of them. Using the definition (14), the equation (21) can be writ- ten in the form: F ( λ ) = F$_{0}$ + λ ∫ 0 dλ ′ λ ′ ∑ − → k σ 1 β ∑ ω n T$_{λ}$ ′ ( k ) × Σ$_{λ}$ ′ ( k ) exp( iω$_{n}$ 0 $^{+}$) . (22) F ( λ ) = F$_{0}$ + λ ∫ 0 dλ ′ ∑ − → k σ 1 β ∑ ω n ϵ ( − → k ) × G$_{λ}$ ′ $_{σ}$( − → k | iω$_{n}$ ) exp( iω$_{n}$ 0 $^{+}$) . (21) − β ∑ − → x − x ′ ∑ σ G$_{λσ}$ ( − → x − − → x $^{′}$|  − 0 $^{+}$) λt ( − → x ′ − − → x ) = − λ ∑ − → k $_{σ}$∑ ω n ϵ ( − → k ) G$_{λσ}$ ( − → k | iω$_{n}$ ) exp( iω$_{n}$ 0 $^{+}$) . (17) ∑ xx ′ G$_{λ}$ ( x | x $^{′}$) λt ( − → x ′ − − → x ) δ ( τ − τ ′ − 0 $^{+}$) δ$_{σσ}$ ′ , (16) 〈 U$_{λ}$ ( β ) 〉 c 0 = − λ ∫ 0 dλ ′ ∑ − → k σω n ϵ ( − → k ) G$_{λ}$ ′ $_{σ}$( − → k | iω$_{n}$ ) exp( iω$_{n}$ 0 $^{+}$) . (20) 〈 U$_{λ}$ ( β ) 〉 c 0 = − ∑ − → x − x ′ βt ( − → x ′ − − → x ) × λ ∫ 0 dλ $^{′}$G$_{λ}$ ′ $_{σ}$( − → x − − → x $^{′}$|  − 0 $^{+}$) . (19) F ( λ ) = F$_{0}$ − 1 β 〈 U$_{λ}$ ( β ) 〉 c $_{0}$, (15) F = F$_{0}$ − 1 β 〈 U ( β ) 〉 c $_{0}$. (14) − ∑ − → x − x ′ ∑ σ β ∫ dλt ( − → x ′ − − → x ) G$_{λσ}$ ( − → x − − → x $^{′}$|  − 0 $^{+}$) . (18)