光子穿过氢分子的旅程是有史以来最短的事件 诸平 The shortest duration ever measured is 247 zeptoseconds — the time it takes for a particle of light, or photon, to get from one side of a hydrogen molecule (illustrated) to another. PETERSCHREIBER.MEDIA/ISTOCK / GETTY IMAGES PLUS 据 玛丽亚·特明 ( Maria Temming ) 2020 年 10 月 27 日发表于《科学新闻》( Science News )的报道,光子从氢分子( H 2 )的一侧(如图示)到达另一侧所需的时间,大约是 247 仄普托秒( about 247 zeptoseconds , 247 zs ),即 247 × 10 -21 s 。这是单个光子穿过氢分子所需的时间,也是有史以来最短的持续时间。相关研究结果于 2020 年 10 月 16 日在《科学》( Science )杂志上发表—— Sven Grundmann,Daniel Trabert,Kilian Fehre,Nico Strenger,Andreas Pier,Leon Kaiser,Max Kircher,Miriam Weller,Sebastian Eckart,Lothar Ph. H. Schmidt,Florian Trinter,Till Jahnke,Markus S. Schöffler,Reinhard Dörner. Zeptosecond birth time delay in molecular photoionization . Science ,16 Oct 2020: Vol. 370, Issue 6514, pp. 339-341. DOI: 10.1126/science.abb9318 研究者在文章指出,这个间隔 大约为 247 zs (1 zs=10 -21 s) 。新的观察结果使物理学家可以在全新的高度上目睹光与物质的相互作用。物理学家将 X 射线光的粒子照在气体中的氢分子上。当每个光子穿过一个 H 2 分子时,它会从一个氢原子踢开电子,然后是另一个氢原子。因为 电子可以表现出波状行为 ( electrons can exhibit wavelike behavior ) ,所以光子与电子的两次撞击事件激发了电子波的扩散和融合,这类似于一粒石子跳过平静的池塘水面两次而形成的波纹。这些波的波峰和波谷重叠形成了 干涉图样 ( interference pattern ),研究人员使用一种称为反应显微镜( reaction microscope )的仪器观察到了该 干涉图样 。 如果同时形成了电子波,则干涉图案将围绕 H 2 分子的中心对称。但是,研究合作者,德国法兰克福歌德大学( Goethe University in Frankfurt, Germany )的物理学家斯文 · 格伦德曼( Sven Grundmann )说,由于一个电子波在另一电子波之前形成,并且有更多的时间扩散,因此波图向第二电子波移动。这一变化使研究人员可以计算出两个电子波发射之间的 247 zs 的时间延迟。根据光速和已知氢分子直径进行计算结果与团队的期望值相吻合。 过去的实验观察到粒子相互作用的时间 短至 阿秒( attoseconds , as )级( as short as attoseconds ), 1 as=10 -18 s, 是 1 zs 的 1000 倍。所以说 这次单个光子穿过氢分子所需的时间是有史以来最短的。 更多信息请注意浏览原文或者相关报道。 The travel time of light in a molecule There is currently considerable interest in experimental studies of various ultrafast processes. Of particular interest are the real-time dynamics of photoionization, one of the most fundamental processes caused by the light-matter interaction, in which the absorption of a photon leads to the ejection of an electron and the formation of anion. Using an electron interferometric technique, Grundmann et al. report a birth time delay on the order of a few hundred zeptoseconds between two electron emissions from the two sides of molecular hydrogen, which is interpreted as the travel time of the photon across the molecule. The proposed technique is generally applicable to more complex systems, and further studies are necessary to support this interpretation. Science , this issue p. 339 Abstract Photoionization is one of the fundamental light-matter interaction processes in which the absorption of a photon launches the escape of an electron. The time scale of this process poses many open questions. Experiments have found time delays in the attosecond (10 −18 seconds) domain between electron ejection from different orbitals, from different electronic bands, or in different directions. Here, we demonstrate that, across a molecular orbital, the electron is not launched at the same time. Rather, the birth time depends on the travel time of the photon across the molecule, which is 247 zeptoseconds (1 zeptosecond = 10 −21 seconds) for the average bond length of molecular hydrogen. Using an electron interferometric technique, we resolve this birth time delay between electron emission from the two centers of the hydrogen molecule.
多原子分子的能级和定态波函数计算 Heitler London 法其实就是 Hartree Fock 方法的双电子情形,详见: https://zhuanlan.zhihu.com/p/40527652 如果说有区别,那就是 HF 后面还有一步变分。 关于 Hartree 以及 Fock 其人: http://blog.sciencenet.cn/blog-365047-496826.html 《量子力学》周世勋: 多原子分子的问题是很复杂的,即使是最简单的双原子分子 -- 氢分子,也只能用近似方法求得。 Heitler-London 法:微扰法,得到结合能为 3.14eV ,实验值为 4.48eV (4.72eV some elsewhere, not sure.) 。 最后得到的基态波函数为: ( 相当于 infinite inter-nuclear separation) 其中,上面的符号是单重态,下面的符号是三重态。 对应的基态能为: 其中各项为: ( 这样 比周世勋的清晰,但 V(1,2) 的定义不太一样,感觉这里有问题) 另外, 1s 的波函数形式,其实周世勋书有详细描述(注意1s=R nl Φ lm ),别的地方一般取 a 0 =1 。 【 slater 行列式】 多电子体系波函数的一种表达方式。 行列式中每一 行 是由 同一电子 的不同可能波函数组成,每一 列 是由 不同电子 的相同可能波函数组成,行列式前的系数是保证波函数归一性的归一系数。 根据行列式的性质,互换行列式中的两行行列式的符号会反转,这一性质 正符合多电子体系的泡利原理 ! 斯莱特行列式在量子化学中应用广泛,经过 自洽场方法 解 HF 方程获得的 最终解 便是一个斯莱特行列式型多电子波函数。 高级的量子化学计算方法也应用到斯莱特行列式, 组态相互作用方法 得到的多电子体系波函数是若干个斯莱特行列式的线性组合,经过对这个由许多行列式组成的巨大波函数的变分法处理,可以获得比 HF 方程更加精确的量子化学计算结果。 slater 行列式命名之有趣历史: http://blog.sciencenet.cn/blog-100379-869543.html 可以从中明白群论在构造多电子波函数中的作用。 【 STO 基组的核心思想】 At finite values of the inter-nuclear distance the exact wave function for hydrogen is no longer of the form of Eq. (1). However , it has been found that Eq. (1) is an excellent approximation to the molecular wave function if the orbitals are adjusted to take into account the effects of molecular formation. 在原始波函数基础上,加入可调节参数,进行优化! 原始波函数一般称为 Slater functions ,因此,该方法就是 Slater-Type-Orbital 基组了。 Dunning, T. H., Hay, P. J. Gaussian basis sets for molecular calculations //Methods of electronic structure theory. Springer, Boston, MA, 1-27 (1977) Make infinite 情况的波函数 Eq. (1) more general ,可以用如下波函数: 该波函数相对于原来的,多了一个参数ζ . 对新的波函数进行优化,得到: 相比原来的 3.14 eV ,更接近 实验值 。 该效果,还可以通过以下的方式达到: 通过优化,可以得到 c 1 =0.663, c 2 =0.351 ,进而 D e =3.74 eV 。 该两种方法考虑的都是 a change in the size of the orbital. 进一步,还可以考虑 polarization effect: 通过在 1s 轨道上引入 2p 轨道的函数: 该形式,允许在 z 方向上有一个不均匀分布,可以达到在成键区域有一个电荷聚集。 通过优化,得到: 更加接近实验值。 【 Gaussian 基组】 1950 年, Boys 提出了可以用高斯形式的函数进行近似: 相比 slater functions ,可能需要 a larger set ,但优化起来更快。 具体形式: 其中,对所有的 s 轨道,都采用 1s 轨道的形式,为 p 轨道采取的形式为: ,有 3 个 d 轨道采取的形式为: 不同个数的 s,p,d 轨道,只是不同的 ζ 。 ζ 的取法为: 这样不管是 s,p 还是 d , ζ 都只有两个参数需要优化。 Hartree Fock 下,发现 9s5p 对 O 原子的描述是比较准确的。 ( 贡献来自于 Huzinaga ) 但计算里德堡态(电子跳到主量子数 +1 的轨道)是不准确的。 使用 augment( 扩展 ) 的,也就是加上 diffusion function ,精确度为提高很多。 ?? 9s5p 是什么意思, 1s,2s, 一直到 9s ?那也太多了吧,不是, 1s 就有 9 个。 First row atom(H , He 是第 0 行 ) 的 negative ions 也不太准确,也需要 augment. 【 contract Gaussian basis set 】 其实,可以对原子,优化得到 9s5p 基组的合适拟合参数,然后作为 contract Gaussian basis set 参与分子的参数拟合。 主要文献: Dunning, Jr., T. H. J. Chem. Phys. 53, 2823 (1970) 以 water molecule 为例, uncontract 的基组为 (9s5p/4s) 。发现 缩并足够好了。但其实 也够了,该基组也被称为 double-zeta, which refers to the use of two basis functions for each atomic orbital. 1977 文献中指出 1s,1s’ 两个只需要一个就够了。这样就成了 。 缩并方案: 1-7/7-8/9--3s 1-4/5--2p 可以看出最外层的 valence atomic orbital 一般是不缩并的,因为其变形比较大。 考虑极化,最佳的基组方案就是 。 for second row atom: (11s7p)/ ,这里以 SO 2 为例进行了比较。 【价分裂基组】 主要发展人 : J. A. Pople 一系列文章: J. Chem. Phys. 51, 2657 (1969) STO-nG 将 Slater function 用 Gaussian function 进行近似,然后再进行 slater function 的优化。指出 STO-3G 是比较合适的。 J. Chem. Phys. 54, 724 (1971) 4-31G 分别指 N 1 =4,N 2 ’=3,N 2 ’’=1 ,见文章,对应与 STO-4g 差不多。 J. Chem. Phys. 56, 2257 (1972) 5-31G , 6-31G Theor. Chim. Acta 28, 213 (1973) 6-31G*, 6-31G** 后续还有 6-311G** ,就用的比较少了: J. Chem. Phys. 72, 650 (1980) 很好的讲解: http://blog.sina.com.cn/s/blog_4822eef90100qmtb.html 赝势基组: d 层轨道原子,如果不对内层进行近似,那么计算量太大。所以,赝势是为了降低计算量,同时保证不降低精度。 对价分裂基组的讲解: https://wenku.baidu.com/view/53bc7e27af45b307e87197df.html 基组分局域的:比如高斯,以及非局域的,比如平面波。 这篇很好,讲解清楚了我(量子化学)和昌英(计算材料学)做的问题的区别: http://blog.sciencenet.cn/blog-365047-839150.html
多国科学家合作首次创建氢分子波函数平方图像Ψ 2 ( H 2 ) 诸平 据物理学家组织网 2018 年 1 月 9 日报道,德国、美国、西班牙、俄罗斯以及澳大利亚的科学家合作首次创建了 氢分子波函数平方图像Ψ 2 ( H 2 ) ,相关研究结果已经于2017年12月22日在《自然通讯》( Nature Communications )网站发表—— M. Waitz, R. Y. Bello, D. Metz, J. Lower, F. Trinter, C. Schober, M. Keiling, U. Lenz, M. Pitzer, K. Mertens, M. Martins, J. Viefhaus, S. Klumpp, T. Weber, L. Ph. H. Schmidt, J. B. Williams, M. S. Schöf fl er, V. V. Serov, A. S. Kheifets, L. Argenti, A. Palacios, F. Martín, T. Jahnke, R. Dörner. Imaging thesquare of the correlated two-electron wave function of a hydrogen molecule . Nature Communications, 2018, 8: 2266 . DOI: 10.1038/s41467-017-02437-9 . H_2 two-electron wave function.pdf 参与此项研究的有来自德国歌德大学( J. W. GoetheUniversität )、德国卡塞尔大学( Universität Kassel )、德国汉堡大学( Universität Hamburg )、德国电子同步加速器中心( Deutsches Elektronen-SynchrotronDESY )、德国 FS-FLASH-D ;西班牙马德里自治大学( Universidad Autónoma de Madrid )、 Instituto Madrileo de EstudiosAvanzados en Nanociencia ;美国劳伦斯伯克利国家实验室( Lawrence BerkeleyNational Laboratory )、美国雷诺内华达大学( University of NevadaReno )、美国中佛罗里达大学( University of CentralFlorida );俄罗斯萨拉托夫州立大学( Saratov StateUniversity );澳大利亚国立大学( The AustralianNational University )的科研人员。审稿人对此评价是具有里程碑意义的研究成果。它不仅是 最前沿的实验结果和非常全面的理论分析的完美结合,而且是氢分子作为一种基本的双电子体系,在相关双电子波函数成像方面的一项原始的、非常有趣的研究,该研究成果结构清晰,同时便于非专业读者阅读。在物理学和化学的广泛领域中,所提出的工作具有很高的影响和激发新思维的潜力。因此,审稿人非常热情地推荐这一具有里程碑意义的作品,在没有任何重大变化的情况下给予发表( Peer Review File )。更多信息请注意浏览原文或者相关报道: Physicistscreate first direct images of the square of the wave function of a hydrogenmolecule January9, 2018 by Lisa Zyga Image of the square of thewave function of a hydrogen molecule with two electrons. Credit: Waitz et al.Published in Nature Communications For the first time, physicistshave developed a method to visually image the entanglement between electrons.As these correlations play a prominent role in determining a molecule's wavefunction—which describes the molecule's quantum state—the researchers then usedthe new method to produce the first images of the square of the two-electronwave function of a hydrogen (H 2 ) molecule. Although numerous techniquesalready exist for imaging the individual electrons of atoms and molecules , this is the first method that can directly image thecorrelations between electrons and allow researchers to explore how theproperties of electrons depend on one another. The researchers, M. Waitz etal., from various institutes in Germany, Spain, the US, Russia, and Australia,have published a paper on the new imaging method in a recent issue of NatureCommunications . There are other methodsthat allow one to reconstruct correlations from different observations;however, to my knowledge, this is the first time that one gets a direct image of correlations by just looking at a spectrum, coauthor FernandoMartín at the Universidad Autónoma de Madrid told Phys.org . Therecorded spectra are identical to the Fourier transforms of the differentpieces of the square of the wave function (or equivalently, to therepresentation of the different pieces of the wave function in momentum space).No reconstruction or filtering or transformation is needed: the spectrumdirectly reflects pieces of the wave function in momentum space. The new method involvescombining two imaging methods that are already widely used: photoelectronimaging and the coincident detection of reaction fragments. The researcherssimultaneously employed both methods by using the first method on one electronto project that electron onto a detector, and using the second method on theother electron to determine how its properties change in response. The simultaneous use of bothmethods reveals how the two electrons are correlated and produces an image ofthe square of the H 2 correlated two-electron wave function. Thephysicists emphasize one important point: that these are images of the squareof the wave function, and not the wave function itself. The wave function is notan observable in quantum physics, so it cannot be observed, Martín said.Only the square of the wave function is an observable (if you have thetools to do it). This is one of the basic principles of quantum physics. Thosewho claim that they are able to observe the wave function are not using theproper language because this is not possible: what they do is to reconstruct itfrom some measured spectra by making some approximations. It can never be adirect observation. The researchers expect thatthe new approach can be used to image molecules with more than two electrons aswell, by detecting the reaction fragments of multiple electrons. The methodcould also lead to the ability to image correlations between the wave functionsof multiple molecules. Obviously, the naturalstep to follow is to try a similar method in more complicated molecules,Martín said. Most likely, the method will work for small molecules, butit is not clear if it will work in very complex molecules. Not because oflimitations in the basic idea, but mainly because of experimental limitations,since coincidence experiments in complex molecules are much more difficult toanalyze due to the many nuclear degrees of freedom. The ability to visualizeelectron-electron correlations and the corresponding molecular wave functions hasfar-reaching implications for understanding the basic properties of matter. Forinstance, one of the most commonly used methods for approximating a wavefunction, called the Hartree-Fock method, does not account forelectron-electron correlations and, as a result, often disagrees withobservations. In addition, electron-electroncorrelations lie at the heart of fascinating quantum effects, such assuperconductivity (when electrical resistance drops to zero at very coldtemperatures) and giant magnetoresistance (when electrical resistance greatlydecreases due to the parallel alignment of the magnetization of nearby magneticlayers). Electron correlations also play a role in the simultaneous emission oftwo electrons from a molecule that has absorbed a single photon, a phenomenoncalled single-photon double ionization. And finally, the results mayalso lead to practical applications, such as the ability to realize correlation imaging withfield-electron lasers and with laser-based X-ray sources. Explore further: Aspace-time sensor for light-matter interactions H_2 two-electron wave function.pdf