请选择 进入手机版 | 继续访问电脑版

计算化学公社

 找回密码
 现在注册!
查看: 1140|回复: 3

[量化理论] 转载:relaxed/unrelaxed density excited state properties

[复制链接]

1万

帖子

25

威望

1万

eV
积分
34891

管理员

公社社长

发表于 2017-5-2 07:19:09 | 显示全部楼层 |阅读模式
Firefly论坛上有一段开发者关于弛豫和非弛豫密度的讨论,说得不错,值得看看

Re: relaxed/unrelaxed density excited state properties  
Alex Granovsky
gran@classic.chem.msu.su

Dear Sanya,
Dealing with the wavefunctions that are the exact solutions
of Schroedinger equation, one can uniquely define one-body
density matrix using pretty simple formulas of Quantum Mechanics.

However, Quantum Chemistry is quite a different case :-)
Here we are forced to use incomplete basis sets and various
approximate methods...

There are computational methods that do have (approximate)
wavefunctions, e.g., HF, CI, MCSCF. Here, to have wavefunction
means that the expressions for energy used within these methods
are exactly equivalent to the expectation value of Hamiltonian
operator calculated using those approximate wavefunctions.

Some other methods do not have (even approximate!) wavefunctions
at all, e.g., DFT; TDHF and TDDFT; MP2, MP3, MP4,...,MPn; CCSD and
any other CC-like methods; EOM-CC etc... This means that the
approximate wavefunction either does not exist at all
(e.g. in DFT, TDHF and TDDFT), or, alternatively, the energy
evaluated within the framework of these methods using some
approximation to "wavefunction", is not equal to the expectation
value of Hamiltonian calculated using this "wavefunction" (e.g.,
all MPn and CC-like methods).

The methods that have approximate wavefunction also have well-defined
expectation-value type one-electron density matrix.
It is defined using standard definition of Quantum Mechanics.
This density matrix has lots of interesting properties, e.g.,
occupation numbers are bracketed between 0 and 2 (for the total
density matrix). The expectation value of any one-electron property
is simply the trace of density matrix multiplied on the operator
representing this property.

However, even in this simplest case, it is possible to define
density matrix of another type - the so-called response-type
density matrix. It is defined so that the derivative of energy
with respect to any one-electron perturbation (x*V) is equal
to the trace of this density matrix times V:

dE
-- = tr(D*V)
dx

These two definitions are identical for exact solutions of Schroedinger
equation. However, for approximate solutions (e.g. working
with incomplete basis sets that is typical for QC) they differs.
More precisely, for those methods where the energy is fully
optimized with respect to molecular orbitals (HF, MCSCF,
complete CI) they are identical, otherwise they are different.
For example, for any incomplete CI (e.g. CISD) one can define both
expectation value and response-type density matrices
. One can
calculate dipole moments using these two density matrices and
get two different answer - the first one is the expectation
value of dipole operator, the second one is the derivative of
CISD energy with respect to the applied electric field.

The response-type density does not have many useful properties of
the "real", expectation value density matrix. For example,
occupation numbers can be arbitrary. However, it does not require
any wavefunction to be defined! One just need to obtain expression
for derivative of total energy with respect to arbitrary one-particle
perturbation - and this can be done for virtually any computational
method, including DFT, MPn, any CC-like, EOM-CC and TD-HF/TD-DFT.
This density matrix is exactly what is usually called "relaxed
density". Typically, it naturally becomes available as the
byproduct during molecular gradients calculations.

Finally, one can define some approximations for density matrix,
even for methods without wavefunction. E.g. for MP2, one can
define approximation to the true density matrix that is correct
up to the second order of PT. For TDHF and TDDFT, one can define
approximation that resembles expectation value density matrix for
CI singles (CIS). This approximation (and there are some good
reasons for this) is what is called "unrelaxed density".

To summarize, one cannot directly compare dipole moments calculated
using relaxed and unrelaxed densities. However, with Firefly, one
can calculate dipole moment as the derivative of energy wrt. to
electric field using runtyp=ffield. This quantity can be directly
compared with dipole moment calculated using relaxed density.

Hope this helps.

Regards,
Alex Granovsky

On Tue Sep 22 '09 6:49pm, sanya wrote
-------------------------------------
>Dear All,

>When calculating excited state properties, such as dipole moments etc., some programs give "relaxed density excited state properties", while others give "unrelaxed density excited state properties". What is the difference between them? As far as I understand, Firefly calculates properties using unrelaxed density (at least, in TDHF and TDDFT). Is it correct to compare, say, dipole moments calculated using relaxed and unrelaxed densities calculated by different programs (with the same functional and basis set, of course)?

北京科音自然科学研究中心:http://www.keinsci.com  致力于计算化学的发展和传播,不定期开办各层次量子化学、分子动力学、波函数分析与Multiwfn程序等主题的培训。欢迎加入“北京科音”微信公众号获取培训最新消息和计算化学资讯
思想家公社的门口Blog:http://sobereva.com(发布大量原创计算化学相关博文)
Multiwfn主页:http://sobereva.com/multiwfn(最流行的量子化学波函数分析程序)
计算化学公社论坛:http://bbs.keinsci.com(高水平、高人气、综合性计算化学交流论坛)
思想家公社QQ群1号:18616395,2号:466017436。用于讨论理论、计算化学,两个群讨论范畴相同,可加入任意其一但不可都加入,申请信息必须注明具体研究方向,否则一概不批。研究方向和理论、计算化学无关者勿加,以免浪费宝贵的空位

此账号为诸Sobereva共用
Money and papers are rubbish, get a real life!

1万

帖子

25

威望

1万

eV
积分
34891

管理员

公社社长

 楼主| 发表于 2017-5-2 07:23:41 | 显示全部楼层
这是更简短的一段话,摘自https://sourceforge.net/p/janpa/wiki/OrcaExamples/

For all non-variational the post-SCF methods (MP2, Coupled Cluster, etc.) two types of electron density (and one-particle reduced density matrix (1-RDM) used to generate natural orbitals) can be defined:

• unrelaxed, or 'expectation value' density (and 'unrelaxed' 1-RDM) which can be calculated by direct integration of approximate wavefunction,

• relaxed, or 'response-type' density (and 'relaxed' 1-RDM) which is a functional derivative of the energy with respect to an external one-particle potential.

A nice discussion by A. Granovskii can be found at Firefly's page Note that construction of relaxed densities requires the solution of so-called 'Z vector equations', but these densities are generally considered to be more appropriate in calculation  molecular properties.

说俗点,弛豫密度就是高斯里做MP2、CCSD、TDDFT等情况时候加了density关键词默认给出的相应级别密度,用的是Z矩阵方法,细节可以看其实现的原文J. Chem. Phys., 81, 5031,好处是只要理论方法能计算一阶导数就能定义相应的密度(但不幸的是,很多高阶理论方法没有一阶解析导数)。而使用比如CIS时,如果用density=rhoci,则波函数直接就是单激发组态函数线性构成的,对应非弛豫密度,此时给出的偶极矩等量相当于在此波函数下相应算符的期望值。

对于TDDFT、CIS等来说,从物理意义上,弛豫密度更接近于电子激发后电子经过弛豫、重排之后达到稳定状态的密度,而非弛豫密度则可以姑且理解为电子激发一瞬间时候的激发态密度。


北京科音自然科学研究中心:http://www.keinsci.com  致力于计算化学的发展和传播,不定期开办各层次量子化学、分子动力学、波函数分析与Multiwfn程序等主题的培训。欢迎加入“北京科音”微信公众号获取培训最新消息和计算化学资讯
思想家公社的门口Blog:http://sobereva.com(发布大量原创计算化学相关博文)
Multiwfn主页:http://sobereva.com/multiwfn(最流行的量子化学波函数分析程序)
计算化学公社论坛:http://bbs.keinsci.com(高水平、高人气、综合性计算化学交流论坛)
思想家公社QQ群1号:18616395,2号:466017436。用于讨论理论、计算化学,两个群讨论范畴相同,可加入任意其一但不可都加入,申请信息必须注明具体研究方向,否则一概不批。研究方向和理论、计算化学无关者勿加,以免浪费宝贵的空位

此账号为诸Sobereva共用
Money and papers are rubbish, get a real life!

22

帖子

0

威望

168

eV
积分
190

Level 3 能力者

发表于 2017-5-2 15:00:42 | 显示全部楼层
简单的说,应该是直接利用收敛波函数或近似波函数计算性质算符的平均值y,如HF,CI,TD等是非弛豫property,如算完HF直接给出的Dipole。如果利用了响应理论,计算微扰加入后能量的一阶梯度,如加入电场计算能量对电场的一阶导数可以得到dipole,是电子密度弛豫后的性质。Dalton中的弛豫的property计算方法最多。

1万

帖子

25

威望

1万

eV
积分
34891

管理员

公社社长

 楼主| 发表于 2017-5-2 19:00:30 | 显示全部楼层
wolfbing 发表于 2017-5-2 15:00
简单的说,应该是直接利用收敛波函数或近似波函数计算性质算符的平均值y,如HF,CI,TD等是非弛豫property ...


HF这种完全变分的波函数,没法说是否是弛豫的,不做这种区分。其能量对电场的导数,和直接用HF波函数算偶极矩算符的期望值是一样的。
截断的CI、TD,都区分弛豫的或非弛豫的密度,高斯里也可以分别给出弛豫和非弛豫的密度,视density关键词的选项而定。
北京科音自然科学研究中心:http://www.keinsci.com  致力于计算化学的发展和传播,不定期开办各层次量子化学、分子动力学、波函数分析与Multiwfn程序等主题的培训。欢迎加入“北京科音”微信公众号获取培训最新消息和计算化学资讯
思想家公社的门口Blog:http://sobereva.com(发布大量原创计算化学相关博文)
Multiwfn主页:http://sobereva.com/multiwfn(最流行的量子化学波函数分析程序)
计算化学公社论坛:http://bbs.keinsci.com(高水平、高人气、综合性计算化学交流论坛)
思想家公社QQ群1号:18616395,2号:466017436。用于讨论理论、计算化学,两个群讨论范畴相同,可加入任意其一但不可都加入,申请信息必须注明具体研究方向,否则一概不批。研究方向和理论、计算化学无关者勿加,以免浪费宝贵的空位

此账号为诸Sobereva共用
Money and papers are rubbish, get a real life!
您需要登录后才可以回帖 登录 | 现在注册!

本版积分规则

手机版|北京科音自然科学研究中心|京公网安备 11010502035419号|计算化学公社 — 北京科音旗下高水平计算化学交流论坛 ( 京ICP备14038949-1号 )

GMT+8, 2018-10-19 18:14 , Processed in 0.154939 second(s), 23 queries .

快速回复 返回顶部 返回列表