{"title":"热力学极限和广义梯度近似中自由电子气体的狄拉克交换能的下阶修正","authors":"Thiago Carvalho Corso, Gero Friesecke","doi":"10.1063/5.0152359","DOIUrl":null,"url":null,"abstract":"We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the local density approximation, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. Generalized gradient approximation (GGA) corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and Perdew, Burke and Ernzerhof we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"539 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations\",\"authors\":\"Thiago Carvalho Corso, Gero Friesecke\",\"doi\":\"10.1063/5.0152359\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the local density approximation, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. Generalized gradient approximation (GGA) corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and Perdew, Burke and Ernzerhof we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.\",\"PeriodicalId\":16174,\"journal\":{\"name\":\"Journal of Mathematical Physics\",\"volume\":\"539 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-08-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0152359\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1063/5.0152359","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations
We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the local density approximation, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. Generalized gradient approximation (GGA) corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and Perdew, Burke and Ernzerhof we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.
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