Multiple domain names and SMTP

Hello,
My question is:
"Do I need to have separate IP addresses for each of my domain names so I can allow SMTP access through the domain names (i.e smtp.domainName1.com, smtp.domainName2.com)?"

Here some issues I'm having:
I have setup multiple domain names on my box successfully.
I have created users and emails addresses with no problem
Users can access their email through a email client like thunderbird.
The problem is they can receive email but they can't send email.
There are 3 domains. Domain number 1's email addresses can send and receive email but the other 2 domains can receive but can't send. The error from the email client is:
"Sending of Message Failed.
The message could not be sent because connecting to the SMTP server smtp.domainname2.com failed. The server maybe unavaiable or is refusing SMTP connections. Please verify that your SMTP server setting is correct and try again, or else contact your network administrator."

The log files show the following when I try to send email:
"
Apr 2 07:57:29 esss000418 postfix/smtpd[2053]: connect from xxx-133-xxx-162.dia.static.qwest.net[xxx.133.xxx.162]
Apr 2 07:57:29 serverName postfix/smtpd[2053]: setting up TLS connection from xxx-133-xxx-162.dia.static.qwest.net[xxx.133.xxx.162]
Apr 2 07:57:30 serverName postfix/smtpd[2053]: Anonymous TLS connection established from xxx-133-xxx-162.dia.static.qwest.net[xxx.133.xxx.162]: TLSv1 with cipher DHE-RSA-AES256-SHA (256/256 bits)
Apr 2 07:57:30 serverName postfix/smtpd[2053]: disconnect from xxx-133-xxx-162.dia.static.qwest.net[xxx.133.xxx.162]
"

Any advise with this would be greatly appreciated.
Thank You

 

ping result of SMTP

Thanks for your reply Till,
I created an A Record entry for smtp at the GoDaddy level and in the ispConfig level. I just have to wait a bit till this takes hold. I'll let you know the results shortly.

As a side note. currently, if I ping smtp.domain2.com shows my machine's ip address.

When I dig smtp.domain2.com it shows:
;; ANSWER SECTION:
smtp.domain2.com. 1529 IN CNAME domain2.com.
domain2.com. 1529 IN A xxx.139.xxx.53

 

SMTP requires authentication

I figured out the issue was I didn't put my username in mail client for smtp authentication.

My Bad.