Partitions of $$N-1$$

Schrödinger term(s) for partition

1 1 1 1

$$\langle VPVPVPVPV\rangle$$

2 1 1

$$-\langle V\rangle \langle V{P}^{2}VPVPV\rangle$$ ;

1 2 1

$$-\langle V\rangle \langle VPV{P}^{2}VPV\rangle$$ ;

1 1 2

$$-\langle V\rangle \langle VPVPV{P}^{2}V\rangle$$

2 2

$${\left(\langle V\rangle \right)}^2\langle V{P}^{2}V{P}^{2}V\rangle$$

3 1

$$-\langle VPV\rangle \langle V{P}^{2}VPV\rangle$$

$${\left(\langle V\rangle \right)}^2\langle VPV{P}^{3}V\rangle$$

1 3

$$-\langle VPV\rangle \langle VPV{P}^{2}V\rangle$$

$${\left(\langle V\rangle \right)}^2\langle VPV{P}^{3}V\rangle$$

4

$$\left[-\langle VPVPV\rangle +\langle V\rangle \langle V{P}^{2}V\rangle \right]\times \langle V{P}^{2}V\rangle$$ ;

$$\langle V\rangle \langle VPV\rangle \langle V{P}^{3}V\rangle$$ ;

$$\langle VPV\rangle \langle V\rangle \langle V{P}^{3}V\rangle$$ ;

$$-{\left(\langle V\rangle \right)}^3\langle V{P}^{4}V\rangle$$

$$S_5=S_1^4+3S_2{S}_1^2+S_2^2+2S_1{S}_3+S_4=1+3+1+4+5=14$$ ;

$$\begin{array}{c}\Delta E_5=\langle VPVPVPVPV\rangle -\Delta E_1\langle V{P}^{2}VPVPV\rangle -\Delta E_1\langle VPV{P}^{2}VPV\rangle \\ -\Delta E_1\langle VPVPV{P}^{2}V\rangle +{\left(\Delta E_1\right)}^2\langle V{P}^{2}V{P}^{2}V\rangle -\Delta E_2\langle V{P}^{2}VPV\rangle \\ +{\left(\Delta E_1\right)}^2\langle V{P}^{3}VPV\rangle -\Delta E_2\langle VPV{P}^{2}V\rangle +{\left(\Delta E_1\right)}^2\langle VPV{P}^{3}V\rangle \\ -\Delta E_3\langle V{P}^{2}V\rangle +2\Delta E_1\Delta E_2\langle V{P}^{3}V\rangle -{\left(\Delta E_1\right)}^3\langle V{P}^{4}V\rangle \end{array}$$