{"title":"Multistate Density Functional Theory: Theory, Methods, and Applications","authors":"Yangyi Lu, Jiali Gao","doi":"10.1002/wcms.70043","DOIUrl":null,"url":null,"abstract":"<p>A quantum theory of density functionals and its applications is presented. By introducing a matrix density <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mfenced>\n <mi>r</mi>\n </mfenced>\n </mrow>\n <annotation>$$ \\mathbf{D}(r) $$</annotation>\n </semantics></math> of rank <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> as the fundamental variable, a one-to-one correspondence has been established between <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mfenced>\n <mi>r</mi>\n </mfenced>\n </mrow>\n <annotation>$$ \\mathbf{D}(r) $$</annotation>\n </semantics></math> and the Hamiltonian matrix representing <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> electronic states—that is, a matrix density functional <span></span><math>\n <semantics>\n <mrow>\n <mi>H</mi>\n <mfenced>\n <mi>D</mi>\n </mfenced>\n </mrow>\n <annotation>$$ \\mathcal{H}\\left[\\mathbf{D}\\right] $$</annotation>\n </semantics></math>. Moreover, no more than <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>N</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$$ {N}^2 $$</annotation>\n </semantics></math> Slater determinants are sufficient to represent <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mfenced>\n <mi>r</mi>\n </mfenced>\n </mrow>\n <annotation>$$ \\mathbf{D}(r) $$</annotation>\n </semantics></math> exactly, giving rise to the concept of minimal active space (MAS). The use of a MAS naturally leads to the definition of correlation matrix functional <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>E</mi>\n <mi>c</mi>\n </msup>\n <mfenced>\n <mi>D</mi>\n </mfenced>\n </mrow>\n <annotation>$$ {\\mathcal{E}}^c\\left[\\mathbf{D}\\right] $$</annotation>\n </semantics></math>, the multi-state extension of the exchange-correlation functional in Kohn–Sham DFT. Variational minimization of the multistate energy, which is defined as the trace of the Hamiltonian matrix functional, yields the exact energies and densities of the lowest <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> eigenstates. A nonorthogonal state interaction (NOSI) algorithm has been developed to optimize the orbitals associated with <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mfenced>\n <mi>r</mi>\n </mfenced>\n </mrow>\n <annotation>$$ \\mathbf{D}(r) $$</annotation>\n </semantics></math> and to approximate the correlation matrix functional. The MSDFT-NOSI method is demonstrated across a range of applications, particularly in cases where KS-DFT and linear-response time-dependent DFT fail, with its accuracy validated through comparison with high-level multiconfigurational wave function theory.</p><p>This article is categorized under:\n\n </p>","PeriodicalId":236,"journal":{"name":"Wiley Interdisciplinary Reviews: Computational Molecular Science","volume":"15 5","pages":""},"PeriodicalIF":27.0000,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://wires.onlinelibrary.wiley.com/doi/epdf/10.1002/wcms.70043","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Wiley Interdisciplinary Reviews: Computational Molecular Science","FirstCategoryId":"92","ListUrlMain":"https://wires.onlinelibrary.wiley.com/doi/10.1002/wcms.70043","RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
A quantum theory of density functionals and its applications is presented. By introducing a matrix density of rank as the fundamental variable, a one-to-one correspondence has been established between and the Hamiltonian matrix representing electronic states—that is, a matrix density functional . Moreover, no more than Slater determinants are sufficient to represent exactly, giving rise to the concept of minimal active space (MAS). The use of a MAS naturally leads to the definition of correlation matrix functional , the multi-state extension of the exchange-correlation functional in Kohn–Sham DFT. Variational minimization of the multistate energy, which is defined as the trace of the Hamiltonian matrix functional, yields the exact energies and densities of the lowest eigenstates. A nonorthogonal state interaction (NOSI) algorithm has been developed to optimize the orbitals associated with and to approximate the correlation matrix functional. The MSDFT-NOSI method is demonstrated across a range of applications, particularly in cases where KS-DFT and linear-response time-dependent DFT fail, with its accuracy validated through comparison with high-level multiconfigurational wave function theory.
提出了密度泛函的量子理论及其应用。通过引入N阶矩阵密度D r $$ \mathbf{D}(r) $$$$ N $$作为基本变量,D r $$ \mathbf{D}(r) $$与表示N $$ N $$电子态的哈密顿矩阵之间建立了一一对应关系,即:矩阵密度泛函H D $$ \mathcal{H}\left[\mathbf{D}\right] $$。此外,不超过n2 $$ {N}^2 $$斯莱特行列式足以精确地表示dr $$ \mathbf{D}(r) $$,由此产生了最小活动空间(MAS)的概念。MAS的使用自然导致相关矩阵泛函E c D $$ {\mathcal{E}}^c\left[\mathbf{D}\right] $$的定义,这是Kohn-Sham DFT中交换相关泛函的多状态扩展。多态能量的变分最小化,定义为哈密顿矩阵函数的轨迹,产生最低N个$$ N $$特征态的精确能量和密度。提出了一种非正交态相互作用(NOSI)算法来优化与D r $$ \mathbf{D}(r) $$相关的轨道并近似相关矩阵泛函。MSDFT-NOSI方法在一系列应用中得到了验证,特别是在KS-DFT和线性响应时变DFT失效的情况下,通过与高级多组态波函数理论的比较,验证了其准确性。本文分类如下:
期刊介绍:
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