General Seniority 学习笔记(1): BCS
我学习了参考文献 [1,2]，把里面的核心公式推导核对整理了，因为觉得有点意思。做完笔记我顺手写了个代码，还没来得及核对。
下一步可以考虑投影 broken-pair 的优化和投影，即 seniority 取次极小的内秉态做投影。
参考文献：
[1] 贾力源，"Application of the variational principle to a coherent-pair condensate: The BCS case", PRC99, 014302 (2019).
[2] 俞一信 et al., "Nucleon pair truncation of the shell model for medium-heavy nuclei", to be submitted.
所有轨道和自己的时间反演轨道，进行两两配对 ((alpha, bar{alpha}))，然后定义时间反演轨道上的配对：
[hat{P}^dagger_alpha = hat{c}^dagger_alpha hat{c}^dagger_bar{alpha}. ]
配对定义为
[hat{P}^dagger = sum_{alpha in Theta} nu_alpha hat{P}^dagger_alpha, ]
其中 (Theta) 表示非集体对序号集合，序号个数是单粒子轨道数的一半。单粒子轨道可以是形变基，也可以是球形基，只要两两配对即可。General Seniority 的全部变化，就是这样的配对的凝聚、对破缺。
一般假定基态对应着配对凝聚，
[|phi_N rangle = frac{1}{ sqrt{chi_N} } (hat{P}^dagger)^N | 0 rangle, ]
其中 (chi_N) 是归一化因子，
[chi_N = langle 0 | hat{P}^N (hat{P}^dagger)^N | 0 rangle. ]
Block 掉非集体对 (alpha) 以后，态矢变为
[|phi^{[alpha]}_N rangle = frac{1}{sqrt{chi^{[alpha]}_N}}( hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^N | 0 rangle, ]
其中，归一化因子
[chi^{[alpha]}_N = langle 0 |hat{P}^N hat{P}_alpha hat{P}^dagger_alpha(hat{P}^dagger)^N | 0 rangle, ]
中间加上 (hat{P}_alpha hat{P}^dagger_alpha) 起的作用就是 block 掉非集体对 (alpha)。
如果已经 block 掉 (gamma_1, gamma_2, cdots, gamma_r)，则有
[chi^{[gamma_1 cdots gamma_r]}_N = N sum_alpha nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}, \ chi^{[gamma_1 cdots gamma_r]}_N - chi^{[alphagamma_1 cdots gamma_r]}_{N} = N^2 nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}. ]
另外，因为 (chi^{[alphagamma_1 cdots gamma_r]}_{0} = 0)，所以可以构造迭代：
[left{chi^{[alphagamma_1 cdots gamma_r]}_{0} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{1} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{1} right} rightarrow cdots rightarrowleft{chi^{[alphagamma_1 cdots gamma_r]}_{N-1} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{N} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right}. ]
也就是说，给定 (gamma_1, cdots, gamma_r)，任意粒子 (N) 的归一化因子都可以如此迭代计算。因为迭代式非常简单，所以预计是秒出。可以定义矢量
[vec{chi}^{[gamma_1, cdots, gamma_r]}_N equiv left( chi^{[gamma_1 cdots gamma_r]}_{N},left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right} right) ]
那么上面的迭代即
[vec{chi}^{[gamma_1, cdots, gamma_r]}_0 = (1,1,cdots, 1), \ vec{chi}^{[gamma_1, cdots, gamma_r]}_{N-1} rightarrow vec{chi}^{[gamma_1, cdots, gamma_r]}_N. ]
test case: (nu_i = 1, i in Theta)
此时 (chi_N) 很容易解析地得到：
[chi_N = C^N_Omega (N!)^2 = frac{Omega!N!}{(Omega-N)!}. ]
通过 many-pair density matrix，可以计算由对算符构造的任意算符矩阵元。
体系哈密顿量为
[hat{H} = sum_{alpha beta} epsilon_{alpha beta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_gamma hat{c}_delta, ]
其中 (V_{alpha betagammadelta} = - langle alpha beta | hat{V} | gamma delta rangle)，有对称性 (epsilon_{alphabeta} = epsilon_{beta alpha}, V_{alphabetagammadelta} = - V_{betaalphagammadelta} = V_{gammadeltaalphabeta})。贾师兄在这里假定了 (epsilon_{alphabeta} = epsilon_{bar{beta}bar{alpha}}, V_{alphabetagammadelta} = V(bar{delta}bar{gamma}bar{beta}bar{alpha}))，据说这是假定 time-even 哈密顿量。他还假定 (epsilon_{alphabeta}) 和 (V_{alphabetagammadelta}) 是实数。
若给定 (gamma_1, cdots, gamma_r)，要求 block 掉 ({ hat{P}^dagger_{gamma_i}}) ，然后在这个截断空间里计算 ((hat{P}^dagger)^{M-p} hat{P}^dagger_{alpha_1} cdots hat{P}^dagger_{alpha_p} |0rangle) 与 ((hat{P}^dagger)^{M-q} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} |0rangle) 的 overlap，
[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} equiv langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} (hat{P}^dagger)^{M-q} |0rangle, ]
这里我们约定 ({alpha_i}, {beta_j})没有共同元素，否则可以移到 ({gamma_i})里面去。这样的话，这么写也可以
[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} (hat{P}^dagger)^{M-q} | 0 rangle, ]
所以可以叫做 many-pair density matrix。容易看出，它可以表示成 (chi) 的表达式：
[t^{[gamma_1, cdots, gamma_r],M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = nu_{alpha_1} cdots nu_{alpha_p} nu_{beta_1} cdots nu_{beta_q} frac{(M-p)!(M-q)!}{(M-p-q)!(M-p-q)!} chi^{[alpha_1 cdots alpha_p beta_1 cdots beta_q gamma_1 cdots gamma_r]}_{M-p-q}. ]
这个比较简单，
[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}_beta (hat{P}^dagger)^N | 0 rangle = delta_{alphabeta} chi_N langle phi_N | hat{n}_alpha | phi_N rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}. ]
只有以下 3 种情况，两体矩阵元不为零：
[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha (hat{P}^dagger)^N |0rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta (hat{P}^dagger)^N | 0 rangle = N^2 nu_alpha nu_beta chi^{[alphabeta]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha (hat{P}^dagger)^N | 0 rangle = N^2(N-1)^2 nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2}. ]
哈密顿量定义为
[hat{H} = sum_{alphabeta} epsilon_{alphabeta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_delta hat{c}_gamma. ]
[langle phi_N | hat{H} | phi_N rangle = sum_{alpha in Theta} 2 epsilon_{alpha alpha} langle phi_N | hat{c}^dagger_alpha hat{c}_alpha | phi_N rangle + sum_{alpha in Theta } V_{alpha bar{alpha} alpha bar{alpha} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha | phi_N rangle \ + sum^{alpha neq beta}_{alpha, beta in Theta} V_{alpha bar{alpha} beta bar{beta} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta | phi_N rangle + sum^{alpha neq beta}_{alpha, beta in Theta} (2V_{alpha beta alpha beta } + 2V_{alpha bar{beta} alpha bar{beta} } ) langle phi_N |hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha | phi_N rangle. ]
将上一小节的结果代入上式，若记 (G_{alphabeta} = V_{alphabar{alpha}betabar{beta}}, Lambda_{alphabeta} = V_{alphabetaalphabeta} + V_{alpha bar{beta} alpha bar{beta} })，得到
[bar{E} equiv langle phi_N | hat{H} | phi_N rangle = frac{N^2}{chi_N} left( sum_{alpha in Theta} (2epsilon_{alphaalpha} + G_{alphaalpha}) nu^2_alpha chi^{[alpha]}_{N-1} + sum^{alpha < beta}_{alpha, beta in Theta} 2G_{alpha beta} nu_alpha nu_beta chi^{[alphabeta]}_{N-1} + (N-1)^2 sum^{alpha < beta}_{alpha, beta in Theta} 2 Lambda_{alphabeta} nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2} right). ]
这个与贾力源的公式一致，只是我的后两个求和符号上有 (alpha < beta)，导致相应的有个 2 倍。最后一项有点 tricky，本就只需考虑 (sum^{alpha<beta}_{alpha,beta in Theta}) 的情况，(V_{alphabetabetaalpha}, V_{alpha bar{beta} alpha bar{beta} }, V_{bar{alpha} bar{beta} bar{alpha} bar{beta} }, V_{bar{alpha} beta bar{alpha} beta })，这四项+时间反演对称，得到 (2Lambda_{alphabeta})。
感觉可以设置 (nu_i = 1)，来做简单测试。
先算 overlap：
[delta chi_N = 2N langle 0 | hat{P}^N hat{P}^dagger_alpha (hat{P}^dagger)^{N-1} |0rangle delta nu_alpha = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1} delta nu_alpha = frac{ 2chi_N}{ nu_alpha} langle phi_N | hat{n}_alpha |phi_Nrangledelta nu_alpha. ]
这可以推广： $ alpha notin { gamma_1, cdots , gamma_r }$ 时，
[delta chi^{[gamma_1 cdots gamma_r]}_N = 2 N^2 nu_{alpha}chi^{[alpha gamma_1 cdots gamma_r]}_{N-1} delta nu_alpha. ]
Anyway, (chi_N) 的偏导数有了
[frac{ partial }{partial nu_alpha} chi_N = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1}. ]
考虑哈密顿量矩阵元的偏导数，
[delta langle hat{P}^N | hat{H} |(hat{P}^dagger)^N rangle = 2N langle hat{P}^{N-1} hat{P}_alpha | hat{H} | (hat{P}^dagger)^N rangle delta nu_alpha， ]
下面计算 (langle hat{P}^{N-1} hat{P}_alpha | hat{H} |(hat{P}^dagger)^N rangle) ，可以如下从容计算。
[N nu_alpha langle (hat{P} - nu_alpha hat{P}_alpha)^{N-1} | hat{H} | (hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^{N-1} rangle = N nu_alpha langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1}. ]
再分别考虑 (hat{H}) 中包含 (alpha) 的各项贡献。
源于 (hat{c}^dagger_alpha hat{c}_alpha, hat{c}^dagger_bar{alpha} hat{c}_bar{alpha}, hat{P}^dagger_alpha hat{P}_alpha)形式的相互作用，与(2epsilon_{alpha alpha} + G_{alpha alpha}) 相关的贡献（对单粒子能）为
[(2epsilon_{alpha alpha} + G_{alphaalpha}) N nu_alpha chi^{[alpha]}_{N-1}; ]源于 (hat{P}^dagger_alpha hat{P}_beta) 形式的相互作用，与 (G_{alpha beta}) 有关的贡献（对散射？）为
[sum_{beta neq alpha} N nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1}; ]源于 (hat{c}^dagger_alpha hat{c}_alpha hat{c}^dagger_beta hat{c}_beta) 形式的相互作用，与 (Lambda_{alphabeta})有关的贡献为
[sum_{beta neq alpha} 2 N(N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta}, ] 这里有个 2 倍，因为相互作用可以是 (hat{c}^dagger_alpha hat{c}_alpha hat{c}^dagger_beta hat{c}_beta)，也可以是 (hat{c}^dagger_beta hat{c}_beta hat{c}^dagger_alpha hat{c}_alpha)。
所以最终得到，
[langle hat{P}^{N-1} hat{P}_alpha | hat{H} |(hat{P}^dagger)^N rangle = N nu_alpha langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) N nu_alpha chi^{[alpha]}_{N-1} \ + sum_{beta neq alpha} N nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 N(N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta}, ]
所以得到了能量偏导数
[ frac{partial}{partial nu_alpha} bar{E} = - frac{1}{chi^2_N} 2N^2 nu_alpha chi^{[alpha]}_{N-1} langle hat{P}^N| hat{H} |(hat{P}^dagger)^N rangle \ + frac{2N^2}{chi_N} left{ nu_alpha langle phi^{[alpha]}_{N-1}| hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) nu_alpha chi^{[alpha]}_{N-1} + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alphanu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} \ = frac{2N^2}{chi_N} left{ - nu_alpha chi^{[alpha]}_{N-1} bar{E} + nu_alpha langle phi^{[alpha]}_{N-1}| hat{H} | phi^{[alpha]}_{N-1}rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) nu_alpha chi^{[alpha]}_{N-1} + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} \ = frac{2N^2}{chi_N} left{ nu_alpha chi^{[alpha]}_{N-1} [ - bar{E} + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle + 2epsilon_{alpha alpha} + G_{alphaalpha} ] + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} ]
有了 (bar{E}) 和 (frac{partial}{partial nu_alpha} bar{E}) 以后，就可以做变分了，改变 (nu_alpha)，指导 (bar{E}) 取得极小值。
为了使得这个式子看起来紧凑一点，贾师兄[1]约定：
[d_alpha = 2 epsilon_{alphaalpha} + G_{alphaalpha} + 2(N-1)^2 sum^{beta neq alpha}_{beta} Lambda_{alphabeta} nu^2_beta frac{ chi^{[alphabeta]}_{N-2}}{chi^{[alpha]}_{N-1}}, ]
那么就有
[nu_alpha = frac{ - sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} }{ chi^{[alpha]}_{N-1} [ d_alpha + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle - bar{E} ] }. ]
然后可以整个迭代式，即 (partial bar{E} / partial nu_alpha = 0)，得到
[nu_alpha = frac{ - sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} }{ chi^{[alpha]}_{N-1} [ d_alpha + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle - bar{E} ] }. ]
上面的公式都是关乎同类核子的，如果哈密顿量中存在质子-中子相互作用，即
[hat{H}_{pinu} = sum_{alpha_pi beta_nu gamma_pi delta_nu} V_{alpha_pi beta_nu gamma_pi delta_nu} hat{c}^dagger_{alpha_pi} hat{c}^dagger_{beta_nu} hat{c}_{delta_nu} hat{c}_{gamma_pi}, ]
对于波函数
[|Phirangle = | phi_{n_pi} rangle |phi_{n_nu} rangle, ]
有
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[langle Phi | hat{H}_{pi nu} |Phirangle = frac{1}{chi_{n_pi} chi_{n_nu}} sum_{alpha_pi beta_nu } (n_pi nu_alpha)^2 (n_nu nu_beta)^2 chi^{[alpha]}_{n_pi -1} chi^{[beta]}_{n_nu -1 } (V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu} + V_{bar{alpha}_pi beta_nu bar{alpha}_pi beta_nu} + V_{bar{alpha}_pi bar{beta}_nu bar{alpha}_pi bar{beta}_nu} ) \ = frac{1}{chi_{n_pi} chi_{n_nu}} sum_{alpha_pi beta_nu } (n_pi nu_alpha)^2 (n_nu nu_beta)^2 chi^{[alpha]}_{n_pi -1} chi^{[beta]}_{n_nu -1 } 2(V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu} ), ]
第二个等号利用了 (V) 的时间反演对称性，即 (V_{alphabetagammadelta} = V_{bar{gamma}bar{delta} bar{beta} bar{alpha}})。
稍微写紧凑点，记 (Lambda_{alpha_pi beta_nu} = 2(V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu}))，利用 ((n_pi nu_alpha)^2 chi^{[alpha]}_{n_pi -1} = chi_{n_pi} - chi^{[alpha]}_{n_pi})，得到[2],
[langle Phi | hat{H}_{pi nu} |Phirangle = sum_{alpha_pi beta_nu } 2 Lambda_{alpha_pi beta_nu} (1 - frac{chi^{[alpha]}_{n_pi}}{chi_{n_pi}}) (1 - frac{chi^{[alpha]}_{n_nu}}{chi_{n_nu}}). ]
这个形式确实好看。
[langle hat{P}^{n_pi-1} hat{P}_alpha; phi_{n_nu} | hat{H}_{pinu} | hat{P}^{n_pi}; phi_{n_nu} rangle = frac{1}{ chi_{n_nu}} sum_{ beta_nu } n_pi nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi -1} (n_nu nu_{beta_nu})^2 chi^{[beta_nu]}_{n_nu -1 } 2Lambda_{alpha_pi beta_nu} \ + frac{1}{ chi_{n_nu}} sum^{gamma_pi neq alpha_pi}_{ gamma_pi beta_nu } n_pi nu_{alpha_pi}(n_pi-1)^2 nu^2_{gamma_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -2} (n_nu nu_beta)^2 chi^{[beta]}_{n_nu -1 } 2 Lambda_{gamma_pi beta_nu} ). ]
其中 (Lambda_{alpha_pi beta_nu} = V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu})，所以有
[frac{partial}{partial nu_alpha} langle Phi | hat{H}_{pi nu} | Phi rangle = frac{1}{ chi_{n_pi} chi_{n_nu}} sum_{ beta_nu } 2 n^2_pi nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi -1} (n_nu nu_{beta_nu})^2 chi^{[beta_nu]}_{n_nu -1 } 2Lambda_{alpha_pi beta_nu} \ + frac{1}{ chi_{n_pi} chi_{n_nu}} sum^{gamma_pi neq alpha_pi}_{ gamma_pi beta_nu } 2n^2_pi nu_{alpha_pi}(n_pi-1)^2 nu^2_{gamma_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -2} (n_nu nu_beta)^2 chi^{[beta]}_{n_nu -1 } 2 Lambda_{gamma_pi beta_nu} \ - frac{2 n^2_pi nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi-1}}{chi_{n_pi}} langle Phi | hat{H}_{pinu} | Phi rangle. ]
用 (37) 式，以及 (frac{ partial }{partial nu_alpha} chi_N = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1}), (frac{ partial }{partial nu_alpha} chi^{[gamma]}_N = 2 N^2 nu_{alpha}chi^{[alphagamma]}_{N-1})，得到
[partial E_{pinu} / partial nu_{alpha_pi} = frac{2n^2_pi nu_alpha}{chi^2_{n_pi}} sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} ( chi^{[gamma]}_{n_pi} chi^{[alpha]}_{n_pi -1} - chi_{n_pi} chi^{[alphagamma]}_{n_pi -1})( 1- frac{ chi^{[beta]}_{n_nu}}{chi_{n_nu}}), ]
(partial E_{pinu} / partial nu_{beta_nu})也可以类似地写出。因此，整个体系的能量 (E = E_pi + E_nu + E_{pinu}) 的偏导数可以写出
[frac{partial E}{partial nu_{alpha_pi} } = frac{partial E_pi}{partial nu_{alpha_pi}} + frac{ partial E_{pinu}}{partial nu_{alpha_pi}} \ = frac{ 2n^2_pi }{ chi_{n_pi} } left{ nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi-1} [ d_{alpha_pi} + langle phi^{[alpha_pi]}_{n_pi-1} | hat{H}_pi | phi^{[alpha_pi]}_{n_pi - 1}rangle - bar{E}_pi ] + sum_{beta_pi neq alpha_pi} nu_{beta_pi} G_{alpha_pi beta_pi} chi^{[alpha_pi beta_pi ]}_{n_pi - 1} + sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} frac{ chi^{[gamma_pi]}_{n_pi} chi^{[alpha_pi]}_{n_pi -1} - chi_{n_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -1}}{chi_{n_pi}}( 1- frac{ chi^{[beta_nu]}_{n_nu}}{chi_{n_nu}}) right} ]
把这个式子做 (pi leftrightarrow nu) 替换，就得到 (partial E / partial nu_{beta_nu}) 的值。
如果整一个迭代式，即 (partial E / partial nu_{alpha_pi} = 0)，得到
[nu_{alpha_pi} = frac{ - sum_{beta_pi neq alpha_pi} nu_{beta_pi} G_{alpha_pi beta_pi} chi^{[alpha_pi beta_pi]}_{n_pi - 1} }{ chi^{[alpha_pi]}_{n_pi - 1} [ d_{alpha_pi} + langle phi^{[alpha_pi]}_{n_pi - 1} | hat{H}_pi | phi^{[alpha_pi]}_{n_pi - 1} rangle - bar{E}_pi ] + sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} frac{ chi^{[gamma_pi]}_{n_pi} chi^{[alpha_pi]}_{n_pi -1} - chi_{n_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -1}}{chi_{n_pi}}( 1- frac{ chi^{[beta_nu]}_{n_nu}}{chi_{n_nu}}) }. ]
这样就可以写开壳核的梯度下降，来做变分了。
General Seniority 学习笔记(1): BCS
我学习了参考文献 [1,2]，把里面的核心公式推导核对整理了，因为觉得有点意思。做完笔记我顺手写了个代码，还没来得及核对。
下一步可以考虑投影 broken-pair 的优化和投影，即 seniority 取次极小的内秉态做投影。
参考文献：
[1] 贾力源，"Application of the variational principle to a coherent-pair condensate: The BCS case", PRC99, 014302 (2019).
[2] 俞一信 et al., "Nucleon pair truncation of the shell model for medium-heavy nuclei", to be submitted.
所有轨道和自己的时间反演轨道，进行两两配对 ((alpha, bar{alpha}))，然后定义时间反演轨道上的配对：
[hat{P}^dagger_alpha = hat{c}^dagger_alpha hat{c}^dagger_bar{alpha}. ]配对定义为
[hat{P}^dagger = sum_{alpha in Theta} nu_alpha hat{P}^dagger_alpha, ]其中 (Theta) 表示非集体对序号集合，序号个数是单粒子轨道数的一半。单粒子轨道可以是形变基，也可以是球形基，只要两两配对即可。General Seniority 的全部变化，就是这样的配对的凝聚、对破缺。
一般假定基态对应着配对凝聚，
[|phi_N rangle = frac{1}{ sqrt{chi_N} } (hat{P}^dagger)^N | 0 rangle, ]其中 (chi_N) 是归一化因子，
[chi_N = langle 0 | hat{P}^N (hat{P}^dagger)^N | 0 rangle. ]Block 掉非集体对 (alpha) 以后，态矢变为
[|phi^{[alpha]}_N rangle = frac{1}{sqrt{chi^{[alpha]}_N}}( hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^N | 0 rangle, ]其中，归一化因子
[chi^{[alpha]}_N = langle 0 |hat{P}^N hat{P}_alpha hat{P}^dagger_alpha(hat{P}^dagger)^N | 0 rangle, ]中间加上 (hat{P}_alpha hat{P}^dagger_alpha) 起的作用就是 block 掉非集体对 (alpha)。
如果已经 block 掉 (gamma_1, gamma_2, cdots, gamma_r)，则有
[chi^{[gamma_1 cdots gamma_r]}_N = N sum_alpha nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}, \ chi^{[gamma_1 cdots gamma_r]}_N - chi^{[alphagamma_1 cdots gamma_r]}_{N} = N^2 nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}. ]另外，因为 (chi^{[alphagamma_1 cdots gamma_r]}_{0} = 0)，所以可以构造迭代：
[left{chi^{[alphagamma_1 cdots gamma_r]}_{0} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{1} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{1} right} rightarrow cdots rightarrowleft{chi^{[alphagamma_1 cdots gamma_r]}_{N-1} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{N} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right}. ]也就是说，给定 (gamma_1, cdots, gamma_r)，任意粒子 (N) 的归一化因子都可以如此迭代计算。因为迭代式非常简单，所以预计是秒出。可以定义矢量
[vec{chi}^{[gamma_1, cdots, gamma_r]}_N equiv left( chi^{[gamma_1 cdots gamma_r]}_{N},left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right} right) ]那么上面的迭代即
[vec{chi}^{[gamma_1, cdots, gamma_r]}_0 = (1,1,cdots, 1), \ vec{chi}^{[gamma_1, cdots, gamma_r]}_{N-1} rightarrow vec{chi}^{[gamma_1, cdots, gamma_r]}_N. ]test case: (nu_i = 1, i in Theta)
此时 (chi_N) 很容易解析地得到：
[chi_N = C^N_Omega (N!)^2 = frac{Omega!N!}{(Omega-N)!}. ]通过 many-pair density matrix，可以计算由对算符构造的任意算符矩阵元。
体系哈密顿量为
[hat{H} = sum_{alpha beta} epsilon_{alpha beta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_gamma hat{c}_delta, ]其中 (V_{alpha betagammadelta} = - langle alpha beta | hat{V} | gamma delta rangle)，有对称性 (epsilon_{alphabeta} = epsilon_{beta alpha}, V_{alphabetagammadelta} = - V_{betaalphagammadelta} = V_{gammadeltaalphabeta})。贾师兄在这里假定了 (epsilon_{alphabeta} = epsilon_{bar{beta}bar{alpha}}, V_{alphabetagammadelta} = V(bar{delta}bar{gamma}bar{beta}bar{alpha}))，据说这是假定 time-even 哈密顿量。他还假定 (epsilon_{alphabeta}) 和 (V_{alphabetagammadelta}) 是实数。
若给定 (gamma_1, cdots, gamma_r)，要求 block 掉 ({ hat{P}^dagger_{gamma_i}}) ，然后在这个截断空间里计算 ((hat{P}^dagger)^{M-p} hat{P}^dagger_{alpha_1} cdots hat{P}^dagger_{alpha_p} |0rangle) 与 ((hat{P}^dagger)^{M-q} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} |0rangle) 的 overlap，
[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} equiv langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} (hat{P}^dagger)^{M-q} |0rangle, ]这里我们约定 ({alpha_i}, {beta_j})没有共同元素，否则可以移到 ({gamma_i})里面去。这样的话，这么写也可以
[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} (hat{P}^dagger)^{M-q} | 0 rangle, ]所以可以叫做 many-pair density matrix。容易看出，它可以表示成 (chi) 的表达式：
[t^{[gamma_1, cdots, gamma_r],M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = nu_{alpha_1} cdots nu_{alpha_p} nu_{beta_1} cdots nu_{beta_q} frac{(M-p)!(M-q)!}{(M-p-q)!(M-p-q)!} chi^{[alpha_1 cdots alpha_p beta_1 cdots beta_q gamma_1 cdots gamma_r]}_{M-p-q}. ]这个比较简单，
[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}_beta (hat{P}^dagger)^N | 0 rangle = delta_{alphabeta} chi_N langle phi_N | hat{n}_alpha | phi_N rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}. ]只有以下 3 种情况，两体矩阵元不为零：
[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha (hat{P}^dagger)^N |0rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta (hat{P}^dagger)^N | 0 rangle = N^2 nu_alpha nu_beta chi^{[alphabeta]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha (hat{P}^dagger)^N | 0 rangle = N^2(N-1)^2 nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2}. ]哈密顿量定义为
[hat{H} = sum_{alphabeta} epsilon_{alphabeta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_delta hat{c}_gamma. ] [langle phi_N | hat{H} | phi_N rangle = sum_{alpha in Theta} 2 epsilon_{alpha alpha} langle phi_N | hat{c}^dagger_alpha hat{c}_alpha | phi_N rangle + sum_{alpha in Theta } V_{alpha bar{alpha} alpha bar{alpha} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha | phi_N rangle \ + sum^{alpha neq beta}_{alpha, beta in Theta} V_{alpha bar{alpha} beta bar{beta} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta | phi_N rangle + sum^{alpha neq beta}_{alpha, beta in Theta} (2V_{alpha beta alpha beta } + 2V_{alpha bar{beta} alpha bar{beta} } ) langle phi_N |hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha | phi_N rangle. ]将上一小节的结果代入上式，若记 (G_{alphabeta} = V_{alphabar{alpha}betabar{beta}}, Lambda_{alphabeta} = V_{alphabetaalphabeta} + V_{alpha bar{beta} alpha bar{beta} })，得到
[bar{E} equiv langle phi_N | hat{H} | phi_N rangle = frac{N^2}{chi_N} left( sum_{alpha in Theta} (2epsilon_{alphaalpha} + G_{alphaalpha}) nu^2_alpha chi^{[alpha]}_{N-1} + sum^{alpha < beta}_{alpha, beta in Theta} 2G_{alpha beta} nu_alpha nu_beta chi^{[alphabeta]}_{N-1} + (N-1)^2 sum^{alpha < beta}_{alpha, beta in Theta} 2 Lambda_{alphabeta} nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2} right). ]这个与贾力源的公式一致，只是我的后两个求和符号上有 (alpha < beta)，导致相应的有个 2 倍。最后一项有点 tricky，本就只需考虑 (sum^{alpha<beta}_{alpha,beta in Theta}) 的情况，(V_{alphabetabetaalpha}, V_{alpha bar{beta} alpha bar{beta} }, V_{bar{alpha} bar{beta} bar{alpha} bar{beta} }, V_{bar{alpha} beta bar{alpha} beta })，这四项+时间反演对称，得到 (2Lambda_{alphabeta})。
感觉可以设置 (nu_i = 1)，来做简单测试。
先算 overlap：
[delta chi_N = 2N langle 0 | hat{P}^N hat{P}^dagger_alpha (hat{P}^dagger)^{N-1} |0rangle delta nu_alpha = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1} delta nu_alpha = frac{ 2chi_N}{ nu_alpha} langle phi_N | hat{n}_alpha |phi_Nrangledelta nu_alpha. ]这可以推广： $ alpha notin { gamma_1, cdots , gamma_r }$ 时，
[delta chi^{[gamma_1 cdots gamma_r]}_N = 2 N^2 nu_{alpha}chi^{[alpha gamma_1 cdots gamma_r]}_{N-1} delta nu_alpha. ]Anyway, (chi_N) 的偏导数有了
[frac{ partial }{partial nu_alpha} chi_N = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1}. ]考虑哈密顿量矩阵元的偏导数，
[delta langle hat{P}^N | hat{H} |(hat{P}^dagger)^N rangle = 2N langle hat{P}^{N-1} hat{P}_alpha | hat{H} | (hat{P}^dagger)^N rangle delta nu_alpha， ]下面计算 (langle hat{P}^{N-1} hat{P}_alpha | hat{H} |(hat{P}^dagger)^N rangle) ，可以如下从容计算。
再分别考虑 (hat{H}) 中包含 (alpha) 的各项贡献。
源于 (hat{c}^dagger_alpha hat{c}_alpha, hat{c}^dagger_bar{alpha} hat{c}_bar{alpha}, hat{P}^dagger_alpha hat{P}_alpha)形式的相互作用，与(2epsilon_{alpha alpha} + G_{alpha alpha}) 相关的贡献（对单粒子能）为
[(2epsilon_{alpha alpha} + G_{alphaalpha}) N nu_alpha chi^{[alpha]}_{N-1}; ]源于 (hat{P}^dagger_alpha hat{P}_beta) 形式的相互作用，与 (G_{alpha beta}) 有关的贡献（对散射？）为
[sum_{beta neq alpha} N nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1}; ]源于 (hat{c}^dagger_alpha hat{c}_alpha hat{c}^dagger_beta hat{c}_beta) 形式的相互作用，与 (Lambda_{alphabeta})有关的贡献为
[sum_{beta neq alpha} 2 N(N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta}, ] 这里有个 2 倍，因为相互作用可以是 (hat{c}^dagger_alpha hat{c}_alpha hat{c}^dagger_beta hat{c}_beta)，也可以是 (hat{c}^dagger_beta hat{c}_beta hat{c}^dagger_alpha hat{c}_alpha)。
所以最终得到，
[langle hat{P}^{N-1} hat{P}_alpha | hat{H} |(hat{P}^dagger)^N rangle = N nu_alpha langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) N nu_alpha chi^{[alpha]}_{N-1} \ + sum_{beta neq alpha} N nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 N(N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta}, ]所以得到了能量偏导数
[ frac{partial}{partial nu_alpha} bar{E} = - frac{1}{chi^2_N} 2N^2 nu_alpha chi^{[alpha]}_{N-1} langle hat{P}^N| hat{H} |(hat{P}^dagger)^N rangle \ + frac{2N^2}{chi_N} left{ nu_alpha langle phi^{[alpha]}_{N-1}| hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) nu_alpha chi^{[alpha]}_{N-1} + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alphanu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} \ = frac{2N^2}{chi_N} left{ - nu_alpha chi^{[alpha]}_{N-1} bar{E} + nu_alpha langle phi^{[alpha]}_{N-1}| hat{H} | phi^{[alpha]}_{N-1}rangle chi^{[alpha]}_{N-1} + (2epsilon_{alpha alpha} + G_{alphaalpha}) nu_alpha chi^{[alpha]}_{N-1} + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} \ = frac{2N^2}{chi_N} left{ nu_alpha chi^{[alpha]}_{N-1} [ - bar{E} + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle + 2epsilon_{alpha alpha} + G_{alphaalpha} ] + sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} + sum_{beta neq alpha} 2 (N-1)^2 nu_alpha nu^2_beta chi^{[alphabeta]}_{N-2} Lambda_{alphabeta} right} ]有了 (bar{E}) 和 (frac{partial}{partial nu_alpha} bar{E}) 以后，就可以做变分了，改变 (nu_alpha)，指导 (bar{E}) 取得极小值。
为了使得这个式子看起来紧凑一点，贾师兄[1]约定：
[d_alpha = 2 epsilon_{alphaalpha} + G_{alphaalpha} + 2(N-1)^2 sum^{beta neq alpha}_{beta} Lambda_{alphabeta} nu^2_beta frac{ chi^{[alphabeta]}_{N-2}}{chi^{[alpha]}_{N-1}}, ]那么就有
[nu_alpha = frac{ - sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} }{ chi^{[alpha]}_{N-1} [ d_alpha + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle - bar{E} ] }. ]然后可以整个迭代式，即 (partial bar{E} / partial nu_alpha = 0)，得到
[nu_alpha = frac{ - sum_{beta neq alpha} nu_beta G_{alphabeta} chi^{[alphabeta]}_{N-1} }{ chi^{[alpha]}_{N-1} [ d_alpha + langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle - bar{E} ] }. ]上面的公式都是关乎同类核子的，如果哈密顿量中存在质子-中子相互作用，即
[hat{H}_{pinu} = sum_{alpha_pi beta_nu gamma_pi delta_nu} V_{alpha_pi beta_nu gamma_pi delta_nu} hat{c}^dagger_{alpha_pi} hat{c}^dagger_{beta_nu} hat{c}_{delta_nu} hat{c}_{gamma_pi}, ]对于波函数
[|Phirangle = | phi_{n_pi} rangle |phi_{n_nu} rangle, ]有
[langle Phi | hat{H}_{pi nu} |Phirangle = frac{1}{chi_{n_pi} chi_{n_nu}} sum_{alpha_pi beta_nu } (n_pi nu_alpha)^2 (n_nu nu_beta)^2 chi^{[alpha]}_{n_pi -1} chi^{[beta]}_{n_nu -1 } (V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu} + V_{bar{alpha}_pi beta_nu bar{alpha}_pi beta_nu} + V_{bar{alpha}_pi bar{beta}_nu bar{alpha}_pi bar{beta}_nu} ) \ = frac{1}{chi_{n_pi} chi_{n_nu}} sum_{alpha_pi beta_nu } (n_pi nu_alpha)^2 (n_nu nu_beta)^2 chi^{[alpha]}_{n_pi -1} chi^{[beta]}_{n_nu -1 } 2(V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu} ), ]第二个等号利用了 (V) 的时间反演对称性，即 (V_{alphabetagammadelta} = V_{bar{gamma}bar{delta} bar{beta} bar{alpha}})。
稍微写紧凑点，记 (Lambda_{alpha_pi beta_nu} = 2(V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu}))，利用 ((n_pi nu_alpha)^2 chi^{[alpha]}_{n_pi -1} = chi_{n_pi} - chi^{[alpha]}_{n_pi})，得到[2],
[langle Phi | hat{H}_{pi nu} |Phirangle = sum_{alpha_pi beta_nu } 2 Lambda_{alpha_pi beta_nu} (1 - frac{chi^{[alpha]}_{n_pi}}{chi_{n_pi}}) (1 - frac{chi^{[alpha]}_{n_nu}}{chi_{n_nu}}). ]这个形式确实好看。
其中 (Lambda_{alpha_pi beta_nu} = V_{alpha_pi beta_nu alpha_pi beta_nu} + V_{alpha_pi bar{beta}_nu alpha_pi bar{beta}_nu})，所以有
[frac{partial}{partial nu_alpha} langle Phi | hat{H}_{pi nu} | Phi rangle = frac{1}{ chi_{n_pi} chi_{n_nu}} sum_{ beta_nu } 2 n^2_pi nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi -1} (n_nu nu_{beta_nu})^2 chi^{[beta_nu]}_{n_nu -1 } 2Lambda_{alpha_pi beta_nu} \ + frac{1}{ chi_{n_pi} chi_{n_nu}} sum^{gamma_pi neq alpha_pi}_{ gamma_pi beta_nu } 2n^2_pi nu_{alpha_pi}(n_pi-1)^2 nu^2_{gamma_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -2} (n_nu nu_beta)^2 chi^{[beta]}_{n_nu -1 } 2 Lambda_{gamma_pi beta_nu} \ - frac{2 n^2_pi nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi-1}}{chi_{n_pi}} langle Phi | hat{H}_{pinu} | Phi rangle. ]用 (37) 式，以及 (frac{ partial }{partial nu_alpha} chi_N = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1}), (frac{ partial }{partial nu_alpha} chi^{[gamma]}_N = 2 N^2 nu_{alpha}chi^{[alphagamma]}_{N-1})，得到
[partial E_{pinu} / partial nu_{alpha_pi} = frac{2n^2_pi nu_alpha}{chi^2_{n_pi}} sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} ( chi^{[gamma]}_{n_pi} chi^{[alpha]}_{n_pi -1} - chi_{n_pi} chi^{[alphagamma]}_{n_pi -1})( 1- frac{ chi^{[beta]}_{n_nu}}{chi_{n_nu}}), ](partial E_{pinu} / partial nu_{beta_nu})也可以类似地写出。因此，整个体系的能量 (E = E_pi + E_nu + E_{pinu}) 的偏导数可以写出
[frac{partial E}{partial nu_{alpha_pi} } = frac{partial E_pi}{partial nu_{alpha_pi}} + frac{ partial E_{pinu}}{partial nu_{alpha_pi}} \ = frac{ 2n^2_pi }{ chi_{n_pi} } left{ nu_{alpha_pi} chi^{[alpha_pi]}_{n_pi-1} [ d_{alpha_pi} + langle phi^{[alpha_pi]}_{n_pi-1} | hat{H}_pi | phi^{[alpha_pi]}_{n_pi - 1}rangle - bar{E}_pi ] + sum_{beta_pi neq alpha_pi} nu_{beta_pi} G_{alpha_pi beta_pi} chi^{[alpha_pi beta_pi ]}_{n_pi - 1} + sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} frac{ chi^{[gamma_pi]}_{n_pi} chi^{[alpha_pi]}_{n_pi -1} - chi_{n_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -1}}{chi_{n_pi}}( 1- frac{ chi^{[beta_nu]}_{n_nu}}{chi_{n_nu}}) right} ]把这个式子做 (pi leftrightarrow nu) 替换，就得到 (partial E / partial nu_{beta_nu}) 的值。
如果整一个迭代式，即 (partial E / partial nu_{alpha_pi} = 0)，得到
[nu_{alpha_pi} = frac{ - sum_{beta_pi neq alpha_pi} nu_{beta_pi} G_{alpha_pi beta_pi} chi^{[alpha_pi beta_pi]}_{n_pi - 1} }{ chi^{[alpha_pi]}_{n_pi - 1} [ d_{alpha_pi} + langle phi^{[alpha_pi]}_{n_pi - 1} | hat{H}_pi | phi^{[alpha_pi]}_{n_pi - 1} rangle - bar{E}_pi ] + sum_{gamma_pi beta_nu} 2 Lambda_{gamma_pi beta_nu} frac{ chi^{[gamma_pi]}_{n_pi} chi^{[alpha_pi]}_{n_pi -1} - chi_{n_pi} chi^{[alpha_pi gamma_pi]}_{n_pi -1}}{chi_{n_pi}}( 1- frac{ chi^{[beta_nu]}_{n_nu}}{chi_{n_nu}}) }. ]这样就可以写开壳核的梯度下降，来做变分了。
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