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DFT的响应理论

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1 / DFT 的 Legendre-transform 结构

HK 定理告诉我们，基态电子密度包含决定基态物理性质所需要的信息。但它本身并没有给出一个一般的构造泛函的方法。根据Levy-Lieb的约束搜索理论，DFT的泛函可以有以下写法

FLL[n]=inf⁡Ψ→n⟨Ψ∣T^+W^∣Ψ⟩.F_{\mathrm{LL}}[n] = \inf_{\Psi\rightarrow n} \langle\Psi|\hat T+\hat W|\Psi\rangle.

而体系的基态能量又是

E[v]=inf⁡Ψ[⟨Ψ∣T^+W^∣Ψ⟩+∫vnΨ].E[v] = \inf_\Psi \left[ \langle\Psi|\hat T+\hat W|\Psi\rangle + \int vn_\Psi \right].

于是直接得到

E[v]=inf⁡n{inf⁡Ψ→n[⟨Ψ∣T^+W^∣Ψ⟩+∫vn]}=inf⁡n{inf⁡Ψ→n⟨Ψ∣T^+W^∣Ψ⟩⏟FLL[n]+∫vn}.\begin{aligned} E[v] &= \inf_n\left\{ \inf_{\Psi\to n} \left[ \langle\Psi|\hat T+\hat W|\Psi\rangle + \int vn \right] \right\}\\[4pt] &= \inf_n \left\{ \underbrace{ \inf_{\Psi\to n} \langle\Psi|\hat T+\hat W|\Psi\rangle }_{F_{\mathrm{LL}}[n]} +\int vn \right\}. \end{aligned}
E[v]=inf⁡n∈IN{FLL[n]+∫v(r)n(r) dr}.E[v]=\inf_{n\in\mathcal I_N} \left\{ F_{\mathrm{LL}}[n] + \int v(\mathbf r)n(\mathbf r)\,d\mathbf r \right\}.

注意这里把搜索空间从波函数空间变成了密度空间。IN\mathcal I_N 是所有允许的、可以由某个有限能量 NN 电子波函数产生的密度集合，其不一定是某个外势下的基态密度。

相当于说 FLL[n]F_{\mathrm{LL}}[n] 可以定义在更大的密度集合上。详见Lieb Density functionals for coulomb systems

把纯态波函数推广为密度矩阵

Γ^=∑kwk∣Ψk⟩⟨Ψk∣\hat\Gamma = \sum_k w_k|\Psi_k\rangle\langle\Psi_k|

对应的混合态（或者系综态）电子密度为

nΓ(r)=Tr⁡[Γ^n^(r)].n_{\Gamma}(\mathbf r) = \operatorname{Tr}\left[ \hat\Gamma\hat n(\mathbf r) \right].

在固定 NN 电子 Hilbert 空间上定义系综泛函

Fens[n]=inf⁡Γ^→nTr⁡[Γ^(T^+W^)].F_{\mathrm{ens}}[n] = \inf_{\hat\Gamma\to n} \operatorname{Tr}\left[ \hat\Gamma(\hat T+\hat W) \right].

纯态只是密度矩阵的特例，因此

Fens[n]≤FLL[n].F_{\mathrm{ens}}[n]\leq F_{\mathrm{LL}}[n].

选取两个允许密度 n1,n2n_1,n_2，其对应的混合态为 Γ1,Γ2\Gamma_1,\Gamma_2。构造

Γλ=λΓ1+(1−λ)Γ2,0≤λ≤1.\Gamma_\lambda = \lambda\Gamma_1+(1-\lambda)\Gamma_2, \qquad 0\leq\lambda\leq1.
nλ=λn1+(1−λ)n2.n_\lambda=\lambda n_1+(1-\lambda)n_2.
Tr⁡[Γλ(T^+W^)]=λTr⁡[Γ1(T^+W^)]+(1−λ)Tr⁡[Γ2(T^+W^)].\operatorname{Tr}[\Gamma_\lambda(\hat T+\hat W)] ={} \lambda\operatorname{Tr}[\Gamma_1(\hat T+\hat W)] +(1-\lambda)\operatorname{Tr}[\Gamma_2(\hat T+\hat W)].

对满足相应密度约束的态取下确界，可以得到

Fens[λn1+(1−λ)n2]≤λFens[n1]+(1−λ)Fens[n2].F_{\mathrm{ens}}[\lambda n_1+(1-\lambda)n_2] \leq \lambda F_{\mathrm{ens}}[n_1] +(1-\lambda)F_{\mathrm{ens}}[n_2].

由此证明系综泛函的凸性。

可以证明将纯态变分空间扩大为所有混合态，并不会降低哈密顿量的全局最低能量。因为混合态能量本质是纯态能量的加权平均，而加权平均不可能低于所有纯态能量的下确界。也即

E0[v]=inf⁡n{FLL[n]+⟨v,n⟩}=inf⁡n{Fens[n]+⟨v,n⟩}.\begin{aligned} E_0[v] &=\inf_n\{F_{\mathrm{LL}}[n]+\langle v,n\rangle\}\\[5pt] &=\inf_n\{F_{\mathrm{ens}}[n]+\langle v,n\rangle\}. \end{aligned}

但采用系综泛函的好处是我们可以利用其凸性实现Legendre-transform

F[n]=sup⁡v{E[v]−∫v(r)n(r) dr}.F[n] = \sup_v\left\{ E[v]-\int v(\mathbf r)n(\mathbf r)\,d\mathbf r \right\}.

即普适泛函F[n]F[n]也可以写成对密度约束搜索的形式。但Legendre-transform不会告诉我们更多信息，其本质只是一个恒等变换。

尝试进行外势的微扰v(r)⟶v(r)+δv(r)v(\mathbf r)\longrightarrow v(\mathbf r)+\delta v(\mathbf r)，哈密顿量的变化为

δH^=∫δv(r)n^(r) dr.\delta\hat H = \int \delta v(\mathbf r)\hat n(\mathbf r)\,d\mathbf r.

根据一阶微扰论

δE=⟨Ψ0∣δH^∣Ψ0⟩.\delta E = \langle\Psi_0|\delta\hat H|\Psi_0\rangle.

于是

δE=∫n0(r)δv(r) dr.\delta E = \int n_0(\mathbf r)\delta v(\mathbf r)\,d\mathbf r.
δE[v]δv(r)=n0(r).\frac{\delta E[v]}{\delta v(\mathbf r)} = n_0(\mathbf r).

再考虑另一种微扰，保持粒子数不变的密度微扰，分析普适泛函F[n]F[n]的变化

∫n(r) dr=N.\int n(\mathbf r)\,d\mathbf r=N.
n(r)⟶n(r)+δn(r)n(\mathbf r)\longrightarrow n(\mathbf r)+\delta n(\mathbf r)

因为 DFT 的基态密度是一个能量最小化问题的解，而在适当的可微性和约束条件下，泛函在极小值处的一阶变分必须为零。当然这一点需要先满足电子数守恒，所以用拉格朗日乘子引入归一化约束

L[n]=F[n]+∫vn−μ(∫n−N).\mathcal L[n] = F[n]+\int vn -\mu\left(\int n-N\right).

有

δLδn(r)=0.\frac{\delta\mathcal L}{\delta n(\mathbf r)}=0.
δF[n]δn(r)=−v(r)+μ.\frac{\delta F[n]}{\delta n(\mathbf r)} = -v(\mathbf r)+\mu.

其中 μ\mu 来自电子数约束。直观理解就是外势加个常数对密度和波函数本身不会有影响。

以上两个式子整理一下可得共轭关系

n(r)=δE[v]δv(r),v(r)=−δF[n]δn(r)+μ.\begin{aligned} n(\mathbf r) &=\frac{\delta E[v]}{\delta v(\mathbf r)},\\[6pt] v(\mathbf r) &=-\frac{\delta F[n]}{\delta n(\mathbf r)}+\mu. \end{aligned}

2 / 响应理论

接着进行更细致的响应分析，考虑一个微小外势扰动

v(r)=v0(r)+δv(r).v(\mathbf r) = v_0(\mathbf r)+\delta v(\mathbf r).

电子密度随之改变

n(r)=n0(r)+δn(r).n(\mathbf r) = n_0(\mathbf r)+\delta n(\mathbf r).

定义静态密度响应函数：

χ(r,r′)=δn[v](r)δv(r′)∣v0.\chi(\mathbf r,\mathbf r') = \left. \frac{\delta n[v](\mathbf r)} {\delta v(\mathbf r')} \right|_{v_0}.

有线性响应关系

δn(r)=∫χ(r,r′)δv(r′) dr′.\delta n(\mathbf r) = \int \chi(\mathbf r,\mathbf r') \delta v(\mathbf r')\,d\mathbf r'.

注意到粒子数守恒，有额外的约束方程

∫δn(r) dr=0.\int\delta n(\mathbf r)\,d\mathbf r=0.
∫dr∫dr′ χ(r,r′)δv(r′)=0.\int d\mathbf r\int d\mathbf r'\, \chi(\mathbf r,\mathbf r')\delta v(\mathbf r')=0.
∫χ(r,r′) dr=0.\int \chi(\mathbf r,\mathbf r')\,d\mathbf r=0.

另外有常数势是静态响应算符的零模。还可以得到（或者可以直接从对称性？）

∫χ(r,r′) dr′=0.\int \chi(\mathbf r,\mathbf r')\,d\mathbf r'=0.

由先前推导的共轭关系，直接推出响应函数就是基态能量泛函对外势的二阶曲率。

χ(r,r′)=δ2E[v]δv(r)δv(r′).\chi(\mathbf r,\mathbf r') = \frac{\delta^2E[v]} {\delta v(\mathbf r)\delta v(\mathbf r')}.

考虑到E[v]E[v]是外势 vv 的凹泛函，如果二阶导数存在，那么自然有

∬u(r)χ(r,r′)u(r′) dr dr′≤0\iint u(\mathbf r)\chi(\mathbf r,\mathbf r') u(\mathbf r')\,d\mathbf r\,d\mathbf r' \leq0

此即响应算符的负半定性。直观理解就是电子密度往往朝着降低外势能量的方向重新分布。另外可以现在可以写出基态能量的展开式

E[v0+δv]=E[v0]+∫n0(r)δv(r) dr+12∬δv(r)χ(r,r′)δv(r′) dr dr′+o(∥δv∥2).\begin{aligned} E[v_0+\delta v] ={}&E[v_0] +\int n_0(\mathbf r)\delta v(\mathbf r)\,d\mathbf r\\ &+\frac12\iint \delta v(\mathbf r) \chi(\mathbf r,\mathbf r') \delta v(\mathbf r')\, d\mathbf r\,d\mathbf r'\\ &+o(\|\delta v\|^2). \end{aligned}

另外再来分析普适泛函的扰动，类似的

δF[n+δn]δn(r)+v(r)+δv(r)=μ+δμ.\frac{\delta F[n+\delta n]}{\delta n(\mathbf r)} +v(\mathbf r)+\delta v(\mathbf r) = \mu+\delta\mu.
∫δ2F[n]δn(r)δn(r′)δn(r′) dr′+δv(r)=δμ.\int \frac{\delta^2F[n]} {\delta n(\mathbf r)\delta n(\mathbf r')} \delta n(\mathbf r')\,d\mathbf r' + \delta v(\mathbf r) = \delta\mu.

令

KF(r,r′)=δ2F[n]δn(r)δn(r′).K_F(\mathbf r,\mathbf r') = \frac{\delta^2F[n]} {\delta n(\mathbf r)\delta n(\mathbf r')}.
KF δn+δv=δμ.K_F\,\delta n+\delta v=\delta\mu.

同样的，这里的δμ\delta\mu可以视作是用来抵消δv\delta v中的常数偏移。当然如果是不包含常数偏移的情况（或者把整个式子投影到不含零模的子空间），也可以直接写

KFδn=−δvK_F\delta n=-\delta v
χ=−KF−1\displaystyle \chi=-K_F^{-1}

以上严格来说其实并未真正用到DFT理论，但显然对无相互作用下的KS体系完全适用，首先列出KS方程

[−12∇2+vs[n](r)]ϕi(r)=εiϕi(r).\left[ -\frac12\nabla^2+v_s[n](\mathbf r) \right]\phi_i(\mathbf r) = \varepsilon_i\phi_i(\mathbf r).
vs[n]=vext+vH[n]+vxc[n].v_s[n] = v_{\mathrm{ext}}+v_H[n]+v_{\mathrm{xc}}[n].
vH[n](r)=∫n(r′)∣r−r′∣ dr′,v_H[n](\mathbf r) = \int \frac{n(\mathbf r')}{|\mathbf r-\mathbf r'|} \,d\mathbf r',
vxc[n](r)=δExc[n]δn(r)v_{\mathrm{xc}}[n](\mathbf r) = \frac{\delta E_{\mathrm{xc}}[n]} {\delta n(\mathbf r)}

假设有效势发生一个小变化导致的轨道变化

vs→vs+δvs.v_s\to v_s+\delta v_s.
ϕi→ϕi+δϕi.\phi_i\to\phi_i+\delta\phi_i.

一阶微扰近似下

∣δϕi⟩=∑a≠i∣ϕa⟩⟨ϕa∣δvs∣ϕi⟩εi−εa.|\delta\phi_i\rangle = \sum_{a\ne i} |\phi_a\rangle \frac{ \langle\phi_a|\delta v_s|\phi_i\rangle }{ \varepsilon_i-\varepsilon_a }.
n(r)=∑ifi∣ϕi(r)∣2,n(\mathbf r)=\sum_i f_i|\phi_i(\mathbf r)|^2,
δn(r)=2Re⁡∑ifiϕi∗(r)δϕi(r).\delta n(\mathbf r) = 2\operatorname{Re} \sum_i f_i \phi_i^*(\mathbf r)\delta\phi_i(\mathbf r).

对比响应函数的展开形式

δn(r)=∫χs(r,r′)δvs(r′) dr′.\delta n(\mathbf r) = \int \chi_s(\mathbf r,\mathbf r') \delta v_s(\mathbf r')\,d\mathbf r'.
χs(r,r′)=−2Re⁡∑i∈occa∈unoccϕi∗(r)ϕa(r)ϕa∗(r′)ϕi(r′)εa−εi.\begin{aligned} \chi_s(\mathbf r,\mathbf r') = -2\operatorname{Re} \sum_{\substack{i\in\mathrm{occ}\\a\in\mathrm{unocc}}} \frac{ \phi_i^*(\mathbf r)\phi_a(\mathbf r) \phi_a^*(\mathbf r')\phi_i(\mathbf r') }{ \varepsilon_a-\varepsilon_i }. \end{aligned}

如果反过来以密度的变化为变量，则注意到Hartree 势和交换关联势的一阶变化是（不需要考虑外势的变化）

δvH(r)=∫δn(r′)∣r−r′∣ dr′.\delta v_H(\mathbf r) = \int \frac{\delta n(\mathbf r')} {|\mathbf r-\mathbf r'|} \,d\mathbf r'.
δvxc(r)=∫fxc(r,r′)δn(r′) dr′.\delta v_{\mathrm{xc}}(\mathbf r) = \int f_{\mathrm{xc}}(\mathbf r,\mathbf r') \delta n(\mathbf r')\,d\mathbf r'.

分别有响应核

fH(r,r′)=1∣r−r′∣.f_H(\mathbf r,\mathbf r') = \frac{1}{|\mathbf r-\mathbf r'|}.
fxc(r,r′)=δ2Exc[n]δn(r)δn(r′).f_{\mathrm{xc}}(\mathbf r,\mathbf r') = \frac{\delta^2E_{\mathrm{xc}}[n]} {\delta n(\mathbf r)\delta n(\mathbf r')}.

定义 fHxc=fH+fxcf_{\mathrm{Hxc}}=f_H+f_{\mathrm{xc}}，于是

δvs=δvext+fHxcδn.\delta v_s = \delta v_{\mathrm{ext}} + f_{\mathrm{Hxc}}\delta n.

另一方面

δn=χsδvs.\delta n=\chi_s\delta v_s.

代入

δn=χsδvext+χsfHxcδn.\delta n = \chi_s\delta v_{\mathrm{ext}} + \chi_s f_{\mathrm{Hxc}}\delta n.

3 / 进一步的微扰分析

References

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