$$N=1$$ : no time points are present on the scale excepting for the beginning-end point; $$S_1=1$$ ;

Schrödinger perturbation energy: $$\Delta E_1=\langle V\rangle$$

$$N=2$$ : only one time point is present on the scale beyond the beginning-end point; $$S_2=1$$ ;

Schrödinger perturbation energy: $$\Delta E_2=\langle VPV\rangle$$ $$N=3$$ , $$N-1=2$$ ;

Partitions of $$N-1$$

Schrödinger term(s) for partition

1 1

$$\langle VPVPV\rangle$$

2

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

$$S_3=S_1^2+S_2=1+1=2$$

$$\Delta E_3=-\Delta E_1^{2}V$$

$$N=4$$ , $$N-1=3$$ ;

Partitions of $$N-1$$

Schrödinger term(s) for partition

1 1 1

$$\langle VPVPVPV\rangle$$

2 1

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

1 2

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

3

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

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

$$S_4=S_1^3+S_2{S}_1+S_1{S}_2+S_3=1+1+1+2=5$$

$$\Delta E_4=\langle VPVPVPV\rangle -\Delta E_1\langle V{P}^{2}VPV\rangle -\Delta E_1\langle VPV{P}^{2}V\rangle -\Delta E_2\langle V{P}^{2}V\rangle +{\left(\Delta E_1\right)}^2\langle V{P}^{3}V\rangle .$$